This little phrase's basic function is to have the last word in an argument. It's dynamic in its meaning, i.e. what it can mean depends on the argument, or it can mean just about anything, regardless of the argument.
Usually, though, it means almost nothing. The "Yeah" part is an almost polite acknowledgement of the other person's last point, and the "but still" implies that somehow the last point you had made was still valid somehow, or that another counterpoint may be forthcoming, and sometimes one is, but oftentimes one isn't. It can get to the point of absurdity whereupon you just say "Yeah, but still" after every subsequent point the other person makes while you're just trying to get the last word.
Most of the time the utterer of YBS is aware of the fact that it's ridiculous and that his or her point or last counterpoint has been trumped quite well and is just saying it out of spite, to just frustrate the other person, or an attempt to be funny, implying that the other person has won but "let's just end this on a little joke."
WARNING: any attempt to counter any point I've made in this little writeup may just be responded to with "Yeah, but still."
Just sayin'.
This could be useful as an insulting softlink in some situations.