And now from Harry's perspective. Harry does at first deny the thought he could be a wizard, until Hagrid pointed out that Harry made things happen he couldn't explain when he was angry or scared, and so many things appeared to fall into place, especially since Petunia confessed she knew all along Harry had to be a wizard due to Lilly being a witch and James being a wizard as well... "How could you not be?" Especially on that moment Harry was technically forced to believe it... It also explained his aunt and uncles rude behavior all those years (since Vernon was merely afraid of Harry's magic, and Petunia was jealous of a gift Harry had that she didn't have, which was also how her relationship with Lily got down the drain).
Now after that Harry has been shopping with Hagrid, and has therefore seen that wizards are quite able to conceal their secrets of Muggle, as Diagon Alley is in the middle of London, but no where to be found unless you know magic (of if a wizard guides you). Given that plus the fact that Harry already saw Percy, Fred and George run through the barrier wall, which conceals the passage to platform 9 3/4, Molly only confirmed once more what Harry had seen before, and thus already knew (even though it was for a short time, but still).
So there's more logic behind it than you initially think....
