The cup will sink if and only if the total downward force pushing on all its upward facing surfaces, plus its own weight, is stronger than the total upward force pushing on its downward facing surfaces.
If the cup is full, then the weight of the water inside it pushes down on the upward facing surface of the inside of the bottom of the cup (and possibly one or more internal side surfaces, if they are at an angle), in addition to the weight of all the air above it.
If the cup is empty then only the weight of the air inside the cup is pushing down on it, in addition again to the weight of all the air above it.
Since water is heaver than air, there is more force pushing down on a cup with water in it than a cup without water in it, and so it is more likely to sink. It's really the same reason that a cup of water feels heavier than a cup of air.