Harry discovers his true nature of wizard when Hagrid comes to see him at the chapter 4 of the first book. Yet, it is a shock to him that Harry doesn't know anything about the magical world he belongs to.
I can't understand how Dumbledore could let it happen. It seems very unlikely he didn't know as I think he was watching over Harry closely, at least to make sure he was safe. Plus, a quote from Hagrid seems to confirm that, as he says that Dumbedore warned him it could be tricky. (In addition to the old cat lady story several books later.)
'I never expected this,' he said, in a low, worried voice. 'I have no idea, when Dumbledore told me there might be trouble gettin' hold of yeh, how mmuch yeh didn't know. Ah, Harry, I don' know if I'm the right person to tell yeh - but someone's gotta - yeh can't go off ter Hogwarts not knowin'.'
Philosopher's Stone, Chapter 4: The Keeper of the Keys
I do know that Dumbledore was thinking it was better for Harry to be far from all the wizards, as he was famous, but it wasn't a reason to let him know nothing about his parents and his true nature.
'It's the best place for him,' said Dumbledore firmly. 'His aunt and uncle will be able to explain everything to him when he's older. I've written them a letter.'
'A letter?' repeated Professor McGonagall faintly, sitting back down on the wall. 'Really, Dumbledore, you think you can explain all this in a letter? These people will never understand him! He'll be famous - a legend - I wouldn't be surprised if today was known as Harry Potter Day in future - there will be books written about Harry - every child in our world will know his name!'
'Exactly,' said Dumbledore, looking very seriously over the top of his half-moon glasses. 'It would be enough to turn any boy's head. Famous before he can walk and talk! Famous for something he wont even remember! Can't you see how much better off he'll be, growing up away from all this until he's ready to take it?'
Philosopher's Stone, Chapter 1: The Boy Who Lived
He never states that it would be wrong to let him know he's a wizard and even says that his aunt and uncle could tell him once he's old enough to understand the news. He didn't want another wizard family to take him in but not that he didn't want Harry to know about his heritage (or even the fact that he is famous).
So my question is how Dumbledore could let Harry be ignorant of this, until he was to go to Hogwarts? I do know many students don't know what a wizard is until they get their letter, but Harry's circumstances is huge and Dumbledore had left a letter more or less for Harry.
As Dumbledore knew that Harry didn't know anything, it seems weird that he didn't go himself to the Dursleys' house to tell Harry everything - as he once did for Tom Riddle - years before the letter to Hogwarts.
(I guess an answer to why Dumbledore didn't came in person would be that he was afraid to mess everything up as he once did with Tom, but it does seem like a bit extrem reasonning.)