Show, don't tell - but what does that actually mean?

Writers who escape from the malign and clammy fingers of the English teacher and take to fiction are always being told that the way to connect with a reader is to 'Show, don't tell'. But what does it actually mean? If you say, for instance:

It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way – in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only.

is this showing or telling? I'll tell you what it is: it's telling. Because the reader is being given a nice big slab of information; you, the writer, are gracing that reader with your opinion and your viewpoint, and nothing's actually happening.
On the other hand, this –

He lay flat on the brown, pine-needled floor of the forest, his chin on his folded arms, and high overhead the wind blew in the tops of the pine trees. The mountainside sloped gently where he lay; but below it was steep and he could see the dark of the oiled road winding through the pass. There was a stream alongside the road and far down the pass he saw a mill beside the stream and the falling water of the dam, white in the summer sunlight.“Is that the mill?” he asked.“Yes.”“I do not remember it.” 

- is showing. But why?
This is why: in the first piece, the beginning of Charles  Dickens' stunning pageturner about the French Revolution, A Tale of Two Cities, Dickens is setting out the mise-en-scène, he's setting up the action first by telling the reader what to think.
In the second piece, Ernest Hemingway's filmic For Whom the Bell Tolls, we jump straight into the action. We have a hero – for Hemingway's characters are always heroes – and a wise old man – for if they're not heroes then they're archetypes – planning to blow up a bridge.
Not that Hemingway's averse to giving you his opinions, but he slithers them into your back pocket under the pretext of telling you what sort of explosives to use, or how thirsty his characters are, or – well, here he is describing his hero:

The young man, who was tall and thin, with sun-streaked fair hair, and a wind- and sun-burned face, who wore the sun-faded flannel shirt, a pair of peasant’s trousers and rope-soled shoes, leaned over, put his arm through one of the leather pack straps and swung the heavy pack up onto his shoulders. He worked his arm through the other strap and settled the weight of the pack against his back. His shirt was still wet from where the pack had rested.“I have it up now,” he said. “How do we go?”“We climb,” Anselmo said.

Very sneaky. But very effective. Not that it's the only only way to write. But it's the only way to write when you're starting to write fiction, because Show, Don't Tell is the magic spell that will free you from the cruel enchantment of those evil English teachers.
When you start to write, you're almost always delighted with all the gorgeous words. Wow! You can use words like 'evanescent'! And 'recursive'!
But you'll go badly wrong if you let the words get you into their net. Because the purpose of writing fiction isn't to show that you've eaten a dictionary. The purpose of writing is to change your reader, by causing an emotional reaction in him, by moving him, by making him see things as you do.
And to do this, you have to get him breathing in synch with you. You have to get the reader feeling that he's experiencing the action.
The best way to do this is to write in action, in scenes. You can give the reader information, but do it in a way that makes the reader control it. I don't know if you remember a film from the 1980s called Diva, about a Vietnamese courier in love with an opera singer who's visiting Paris? In the middle of an intensely complicated and arty and très très French series of twists and turns, the villains corner one of our heroes in a friend's warehouse, where he's hiding out.
We've met this friend, Gorodish, before – when we first meet him, he's chopping onions, wearing a snorkel and a mask to protect himself against the tear-inducing fumes. Later, as one of these same villains is smirkingly about to kill someone, Gorodish appears from behind him, cracks an ampoule of something magical, and the villain's eyes turn up and he folds up into unconsciousness.
Now, Gorodish reappears in the dark, and quietly swings the switch that controls the lights so that it's over the un-railed-off hole where the industrial lift comes up and down. The switch has a red bulb lighting it up so you can see it in the dark.
Teeth gleaming in anticipation, the villain reaches over for the switch, and "Aaaaaaaarrrrgggghhhhhh!!" disappears, and there's a crash below.
Now, that's good writing.
What do you remember from the books you love most? Almost invariably, it's a scene. You may love it for what it portends, like this scene from Pádraic Colum's The King of Ireland's Son –

"Prince," said the old fellow looking up at him, "if you can play a game as well as you can sing a song, I'd like if you would sit down beside me.""I can play any game," said the King of Ireland's Son. He fastened his horse to the branch of a tree and sat down on the heap of stones beside the old man."What shall we play for?" said the gray old fellow."Whatever you like," said the King of Ireland's Son."If I win you must give me anything I ask, and if you win I shall give you anything you ask. Will you agree to that?""If it is agreeable to you it is agreeable to me," said the King of Ireland's Son.

- when you immediately know that this easygoing young lad, who'll go a bit of the road with anyone, is about to get himself into more trouble than he can easily get out of.
Foreshadowing like this is a lovely compliment to the reader: the writer gives the reader a bit of information that's going to come in handy, and the reader is delighted to guess what's likely to happen.
Colum could, of course, have simply told the reader:

"Now, little did the King of Ireland's Son know it, but the man who was asking to play a game of chess with him was a dangerous enchanter"

– and a less skilled writer would have done so. That would be telling.
Instead, he chose to show the unfortunate young lad as his inattention and sloppy good nature topples him from the indulged life of a spoilt boy into all the difficulties and sorrows of an adventurer. And because he showed it, the reader followed eagerly after.
What's your favourite book? What do you remember from it? You may, for instance, remember Jane Austen's famous first line from the comic romance Pride and Prejudice –

It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife.

- but you only remember this nice piece of cattiness because of the action that starts almost immediately -

"My dear Mr Bennet,'' said his lady to him one day, "have you heard that Netherfield Park is let at last?''Mr Bennet replied that he had not."But it is,'' returned she; "for Mrs Long has just been here, and she told me all about it.''Mr Bennet made no answer."Do not you want to know who has taken it?'' cried his wife impatiently."You want to tell me, and I have no objection to hearing it.''This was invitation enough.

And what you remember from the story that follows, with all its satisfying misunderstandings and star-crossings, is almost certainly not Austen's superb prose, the epitome of a well-informed maiden aunt's murmurings, but the action: Mrs Bennet's efforts to get her daughters married to wealthy Mr Bingley and other good prospects; sensible, ethical Elizabeth's dislike of high-nosed Mr Darcy, the schemings of the seducer Wickham, and Elizabeth's turnaround when she discovers that Mr Darcy has secretly paid Wickham's debts so that he can marry… well, you know.
What do you remember of A Tale of Two Cities? Certainly most of the rest is lost to you, while that scene where the worthless Sidney Carton changes place with his heroic lookalike Charles Darney and steps onto the platform of the guillotine with the words "It is a far, far better thing I do than I have ever done…" to save his life and give him to the woman Carton loves.
What do you remember of The Call of the Wild? Every word of Jack London's book is action – and you'll remember the scenes where the dog Buck is rescued after long episodes of brutality, when Buck saves his rescuer from a river, and the end, where Buck, now free, returns to mourn his master.
Romeo and Juliet? There's plenty of resonant language, which you remember because it's woven in with the emotions: if you're asked to recall any scene, it'll be the balcony scene and the death of the two lovers through misunderstandings.
Every good writer does this: brings the story into the consciousness of the reader. Look at Chekhov starting a story:

It was said that a new person had appeared on the sea-front: a lady with a little dog. Dmitri Dmitritch Gurov, who had by then been a fortnight at Yalta, and so was fairly at home there, had begun to take an interest in new arrivals. Sitting in Verney's pavilion, he saw, walking on the sea-front, a fair-haired young lady of medium height, wearing a béret; a white Pomeranian dog was running behind her.And afterwards he met her in the public gardens and in the square several times a day. She was walking alone, always wearing the same béret, and always with the same white dog; no one knew who she was, and every one called her simply "the lady with the dog.""If she is here alone without a husband or friends, it wouldn't be amiss to make her acquaintance," Gurov reflected.

You know immediately what kind of person Gurov is, from just this short passage. Even when Chekhov goes on to give you some information about the character – 

He was under forty, but he had a daughter already twelve years old, and two sons at school. He had been married young, when he was a student in his second year, and by now his wife seemed half as old again as he. She was a tall, erect woman with dark eyebrows, staid and dignified, and, as she said of herself, intellectual. She read a great deal, used phonetic spelling, called her husband, not Dmitri, but Dimitri, and he secretly considered her unintelligent, narrow, inelegant, was afraid of her, and did not like to be at home. He had begun being unfaithful to her long ago -- had been unfaithful to her often, and, probably on that account, almost always spoke ill of women, and when they were talked about in his presence, used to call them "the lower race."

- he does it in such a way that it's causing you, the reader, to form your own opinions, slyly making you feel a sharp dislike of Mr Gurov.
Even in Tipperary man Laurence Sterne's hilarious 18th-century romp through France, A Sentimental Journey -

They order, said I, this matter better in France. - You have been in France? said my gentleman, turning quick upon me, with the most civil triumph in the world. - Strange! quoth I, debating the matter with myself, That one and twenty miles sailing, for ’tis absolutely no further from Dover to Calais, should give a man these rights: - I’ll look into them: so, giving up the argument, - I went straight to my lodgings, put up half a dozen shirts and a black pair of silk breeches, - “the coat I have on,” said I, looking at the sleeve, “will do;” - took a place in the Dover stage; and the packet sailing at nine the next morning, - by three I had got sat down to my dinner upon a fricaseed chicken, so incontestably in France, that had I died that night of an indigestion, the whole world could not have suspended the effects of the droits d’aubaine; - my shirts, and black pair of silk breeches, - portmanteau and all, must have gone to the King of France; - even the little picture which I have so long worn, and so often have told thee, Eliza, I would carry with me into my grave, would have been torn from my neck! - Ungenerous! to seize upon the wreck of an unwary passenger, whom your subjects had beckoned to their coast! - By heaven!  Sire, it is not well done; and much does it grieve me, ’tis the monarch of a people so civilized and courteous, and so renowned for sentiment and fine feelings, that I have to reason with! -

while Sterne is chattering like the chattiest friend you've ever had, you're getting lots of info from the music of what's happening, from what you're being shown, not told.