Louis knows that he was never supposed to have the diary in the first place. It's private, and most of all, it's Harry's, but he loves him. That's how he justifies it. He loves Harry, so it's okay to know all these things. And that's true, except for the fact that Harry hasn't told Louis about what he's written in his diary because there are things that are hard to relive.
The thing with Louis is that he always starts off with good intentions, but things don't always go as planned. The original plan was just a quick glance and then find a way to give the book back to Harry, but Louis is still the curious person who fell in love with not only a boy, but the thought of solving a big mystery about his past.
Now Louis is hooked and he can't figure out how to put the diary back. Especially since Harry is starting to realize it's missing, but he still trusts Louis and maybe he's a little naive.
Things are going downhill fast, all because Louis is too nosy. And you know what they say: curiosity killed the cat.