First off, The Curious Incident of the Dog in the Nighttime was an amazing book, just saying.
Second, I'll give the answer a go!
Christopher's trip to London helped make him feel more independent and to show him that he could do things on his own. He was able to go all on his own with no help (except from his rat, Toby, of course). And so, this trip showed him that he was even smarter than he thought he was.
This is not the full answer. If you need more, maybe go back over the book a little bit!
I hope this helps out!