It means that if you press X when you see that, your last race or battle will be recorded. You can save up to six plays and watch them again on the MKTV section. MKTV automatically records your last (I believe) twelve races/battles, but they cycle out. If you mark it as a favorite, it stays, instead of getting deleted. If it says "1/6" that means "this will be #1 of your 6 available play saves".