now that I reread the question again I am not sure if it is why light bulbs break or why light bulbs break only when the current is on. Anyway, I found this:
Tungsten gains resistance as it heats. Resistance is the amount of ‘push back' a material has against an electric current. The only thing that heats tungsten in a light bulb is electric current flowing through it. Imagine if a rubber hose gained strength only after water flowed through it. After some use, it would be able to handle a heavy stream of water. At first, though, it would bulge and strain like a water balloon before regaining its shape. Unless the rubber is in good condition, it will snap. In the same way, a tungsten filament is overloaded with current in the first few seconds after being turned on. The heat causes it to expand, and the filament experiences thermal stress, the strain of the material trying to expand due to sudden changes in temperature. Unless it is in good condition, it snaps.