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In Harry Potter and the Chamber of Secrets, Uncle Vernon says:

We even let you have Dudley's second bedroom purely out of the goodness of our hearts.

There is plenty of evidence that there's no "goodness in their hearts" when it comes to Harry. They had no problem with him living under the stairs for almost 11 years. Why did that change?

Did someone in the wizarding world force them?

Did they decide to use the cupboard under the stairs for something else?

Or (much less likely), did they develop a tiny bit of sympathy for Harry?

The Dursleys moved him because they were worried that Dumbledore (or whoever was sending them Harry's acceptance letters) had made it clear that they were aware that Harry was being kept in a cupboard.

Mr H. Potter
The Cupboard under the Stairs

4 Privet Drive

Little Whinging

Surrey

_{^{Harry Potter and the Philosopher's Stone}}

Shortly after they moved him, the letters reflected the change, even if they were still a bit sarcastic.

‘Er – yes, Harry – about this cupboard. Your aunt and I have been thinking … you’re really getting a bit big for it … we think it might be nice if you moved into Dudley’s second bedroom.’

‘Why?’ said Harry.

‘Don’t ask questions!’ snapped his uncle. ‘Take this stuff upstairs, now.’

and

‘There’s another one! Mr H. Potter, The Smallest Bedroom, 4 Privet Drive –’

If we're talking specifically about the films, there was also an additional exposition scene in the original Harry Potter and the Philosopher's Stone screenplay that was written but not filmed that shows how concerned his aunt and uncle are about the fact that the letter-sender can see where he sleeps.

They gave it to him in the first book, after the incident with the letters.. but as to keeping it, I don't remember it being expressly stated, but I believe it was due to them being afraid of what he might do with his magic, once he learned he had it, and how to use it.

From the same book:

Underage wizards weren’t allowed to use magic outside of school. Harry hadn’t told the Dursleys this; he knew it was only their terror that he might turn them all into dung beetles that stopped them from locking him in the cupboard under the stairs with his wand and broomstick.

(Emphasis added)

Since that was where he stayed before, it seems pretty consistent with that being the reason.

In the first book, near the end, we see:

“Hope you have—er—a good holiday,” said Hermione, looking uncertainly after Uncle Vernon, shocked that anyone could be so unpleasant.

“Oh, I will,” said Harry, and they were surprised at the grin that was spreading over his face. “They don’t know we’re not allowed to use magic at home. I’m going to have a lot of fun with Dudley this summer…”

Beyond that, I think they were somewhat afraid of what the Wizardling World might do in response to their 'Hero' being abused.. That didn't result in good treatment, but at least the over-the-top 'keep him under the stairs' issue went away. Overall, a lesser version of the protection he gets at the end of Prisoner.

“It’s not,” said Harry cheerfully. “It’s a letter from my godfather.”

“Godfather?” sputtered Uncle Vernon. “You haven’t got a godfather!”

“Yes, I have,” said Harry brightly. “He was my mum and dad’s best friend. He’s a convicted murderer, but he’s broken out of wizard prison and he’s on the run. He likes to keep in touch with me, though… keep up with my news… check if I’m happy…”

And, grinning broadly at the look of horror on Uncle Vernon’s face, Harry set off toward the station exit, Hedwig rattling along in front of him, for what looked like a much better summer than the last.