"Stick around" means to stay. It is a more casual way to describe staying somewhere. You can use it when talking to people in social situations:
Hey, why don't you stick around and have dinner with us?
I'd love to stick around, but I have to go pick up the kids from soccer practice.
Or you can use it in professional situations if you want to sound casual:
Hey, would you mind sticking around for a bit after the meeting? There's something I want to run by you.