To earn your belts, you must play in Competition Mode and defeat many opponents. As you keep earning more belts, you have to start defeating more people. There are a total of nine belts that you can earn. Like I said, you earn a belt after you win a certain amount of times. Here is a chart that shows you how many wins you need to earn your next belt

White belt - 5 total wins needed.
Yellow belt - 13 total wins needed.
Orange belt - 21 total wins needed.
Green belt - 30 total wins needed.
Blue belt - 40 total wins needed.
Red belt - 52 total wins needed.
Purple belt - 64 total wins needed.
Brown belt - 76 total wins needed.
Black belt - 88 total wins needed.