Ice cubes melt, but as they come into contact with other ice they start to freeze again, but this time they freeze as one.
Ice is cold, that is the primary reason why we use them. The ice is put in a glass of liquid with the purpose of cooling the liquid; hence it can be said that the temperature of ice is less than the temperature of the liquid.
When the ice is put into the liquid, the pressure changes, i.e. the pressure of air is different than the pressure of a liquid. This pressure is actually what leads to the temperature difference. The internal pressure of the ice cube is different than the external pressure applied on it from all sides. This external pressure leads to the ice melting.
However, as the ice is melting there are chances that the ice cube will come into contact with another ice cube in the glass. When this happens, the pressure pushes the ice cubes together. The internal pressure of the ice causes the ice cubes to refreeze. While refreezing, they tend to also freeze the little liquid there is between the ice cubes. This liquid acts as a bridge between the two cubes and links them together until they actually become one big ice cube.
However, they don’t truly become one big ice cube, as they can easily be snapped apart. Still, it gets harder and harder to snap them apart the longer they are stuck together. This is because the longer they are in contact the stronger the bridge becomes.
Note that ice that is at freezing temperature, i.e. 0 deg C does not stick together with other. Where else, ice that is below freezing temperature does not. Also, if the ice is allowed to come up to freezing temperature before it is allowed to touch another ice cube, it will not clump together. This can be done by mixing the ice and the liquid in a glass together, and stirring it continuously for a couple of minutes until the temperature of the ice has increased.