The difference between "trovarsi" ans "stare": both mean "I am staying in London". Literally, "trovarsi" means "I find myself to be", "stare" is more like the chinese dai4.
"Trovarsi" is a more classy way of speaking, you can use it in business conversations, or if you're talking to your mother-in-law, "stare" is more colloquial, very widely used.
Getting the hang of "usage" is difficult, just keep asking questions.