When some who doesn't have a place to stay, lives with you for three weeks and then floods your apartment.
Hey Travis, I know you do not have a place to stay and you been here for three weeks but you just flooded my apartment. You just showed me what "wear out your welcome" means so GET OUT!!!!
This phrase has two meanings; the first is that you're mother gave you a badass name as a baby and you've grown up with an immense ego about it, so you don't want people to ruin it with their nasty voices. The other meaning is that you don't care about people so you avoid them bugging you, by saying this phrase.
For example if your name is CandiCorona, you have all the right in the world to say "That's my name, don't wear it out.