Why do ice cubes stick together in a glass?

Pressure causes ice to melt. Actually, this is what causes an iceberg/glacier to move. The tremendous weight of the glacier causes the ice on the very bottom to melt, creating a layer of water on which the glacier glides.

When two ice cubes in a glass of water touch for a prolonged period of time, the slight pressure of one against the other causes a little bit of ice at the point of contact to melt. The melted water then flows away from the point of contact and immediately refreezes. This refrozen water forms sort of a "weld" which causes ice cubes to stick together.