Look at it this way:
Driving your car isn't a constitutional right.
However, if the government allows people to drive cars, they cannot limit it in an unconstitutional way. They cannot say "all men can drive cars but women can't" or "only whites may drive cars."
So the argument is this: Being gay should be one of the unconstitutional prohibitions under the 14th amendment. We want all laws to never have a provision that says "Everyone can drive a car (or get married, or be in the army etc.) except gays."