Hi zacsee2000,
You have to describe in a slightly greater detail as to how you pour the glass of ice water. Once you are saying that the cubes stick together, but then you say that they melt when they touch each other. Do they also stick together side by side? Do you put the water afterwards?
Defennder has interestingly explained it through regelation, but I doubt if a single ice cube will exert too much of a pressure on a lower one to cause sufficient melting and re-freezing .
Andy's explanation seems better because the temperature of the ice cubes when you take them out of the icebox (-20 C) is much lower than the room temperature. Water will definitely condense on the boundary, but are the ice cubes cold enough to "freeze" that water again?
So, a detailed description of what happens when you put them together without water would be helpful. (In the meantime, I can see myself taking too much of iced beer -- well, anything for a good cause...)