“I've got to get my own place.”

You live with your parents, but you get really annoyed with them. You're fed up with it and feel like you really want to move into an apartment by yourself.

I've got to get my own place.

I've got to (do something)

When you really want to do something and feel like you can't wait any longer, you say "I've got to ___":

I've got to get an iPad.

This is pronounced with stress on the word "got".

In casual speech, "got to" can be pronounced "gotta". In that case, the stress falls on the first syllable, "gotta".

(one's) own (something)

In this case, "my own" means something that you don't share with other people. So your own apartment would be one that you live in by yourself.

(one's) place

"My place" is a casual way to refer to your house or apartment. It's usually used by young people, and almost always in spoken conversation.