What is the meaning of this quote?
"Running your mouth off" is another way of saying "talking too much", usually about the wrong things at the wrong time. So the quote is saying if you run your mouth off all the time, chances are that you're going to meet someone who can't put up with it, and they will make you stop talking.
If you are not careful in what you say someday someone will make you pay
It means one day you will offend someone and they will hurt you.
It means if you keep on being mouthy you are going to upset the wrong person who will knock you out/haul you into court/give you the sack.
Watch your mouth or you'll be sorry.
YOUR. Not YOU'RE. YOU'RE = you are This is not rocket science.
It means whoever wrote it doesn't know the difference between "your" and "you're".