It depends on the context! But most of the times you'll hear this when in a situation like this: (I'm going to use an example) Let's say you had tried to book an appointment with a Doctor, but he replied he wasn't available on that day, he was very busy! So you tell him it's very important and insists that he tries to find some space in his schedule, after all it's very important! He then checks his schedule up another time finds some free time and FITS you in his busy schedule, now you can go to the doctor on that day! I hope I made myself clear. To fit someone in is an expression used when someone can accommodate another person in a space or time that wasn't clearly available at first.

You would say this to someone when they make room in their busy schedule to meet with you, usually for an appointment. You might also hear people say: "Thanks for squeezing me in." :)

It means "Thanks for finding time for me today", it also depends on the context a bit.