One interpretation is that the marriage will last as long as the glass is broken—forever. Other more superstitious say a loud noise is thought to drive away evil spirits. Another reason given was that this is a reminder that although the couple came together as a single union, the world as a whole is broken and needs mending. Some have said that it is a reminder that even in times of great joy that there is sadness. That life will bring sadness as well as joy.
Most however do seem to agree that the Jewish Wedding and the breaking of the glass is a reminder of the destruction of the Temple in Jerusalem. This act serves as an expression of sadness at the destruction of the Temple in Jerusalem. A Jew, even at the moment of greatest rejoicing, should be mindful of the Psalmist's injunction to set Jerusalem above their highest joy.
There is this strangely specific little article:
http://www.dangoor.com/75059.html
... though as someone from Germany (in my specific case, the rural Catholic south, though I think the following is true throughout Germany), I will say that for as long as I can remember, it has been common tradition to have a "Polterabend" (literally something like "rambunctious eve") on the night before the wedding, a central part of which is the breaking of earthenware or porcelain (most weddings I've been to, they threw a washbasin to the ground, though not seldom a toilet bowl is used (newly bought at a DIY store of course...); the soon-to-be-wed couple then cleans up jointly with a broom.
Not sure whether and how that relates to the specifically Judaic custom, though what the other commenters said makes clearly sense: i.e., that these things probably form a part of human universals:
making a vanitas reminder of fragility and impermanence an integral part of rituals that mark a watershed in people's lives;
"ward off evil spirits" (even if one is a complete agnostic) with a symbolic knocking-on-wood gesture;
at joyous events: include a debbie-downer safeguard so one is not "too joyous".
Breaking something (or letting fire consume something, or burying something) that would otherwise still be useful expresses all these ideas nicely, and is thus something many cultures would converge upon, and indeed do.