Why does the mass of ice increase when it melts?

It doesn't. If you have 1 kg of ice, it becomes 1 kg of liquid water. If it became more massive that would mean there were more particles appearing from nowhere, which is of course impossible. What does increase is the density, i.e. the mass of 1 metre cubed. If you start with 1 metre cubed of ice, it will form a smaller volume when it melts (but it will still have the same mass). The reason it becomes more dense is that in ice the molecules arrange themselves in a particular way so that they produce a very open structure, and when it melts they actually move closer together, because they don't have to line up in that way.