Why do frozen objects have a higher propensity to snap/break cleanly than non-frozen/warmer ones?

First of all, it is not always the case -- the most known example is meat frozen in liquid nitrogen. In general, I think there are many different processes involved; temperature change may lead to small fractures in solid materials, in organic objects we have formation of ice from water inside cells what both damages the structure and lowers elasticity, there may be more general phase transitions involved, like for instance crystal structure changes of cocoa butter in chocolate.

Depends on the kind of object.

A really "hard" object (i.e. a crystal) will break just as easy when it's warm, but most other materials are flexible and/or soft, so they give way before breaking. Cooling them lowers that flexibility, so e.g. a cold string of rubber will snap much earlier, but a string of crystal will have snapped when stretched that far more or less regardless of it's temperature.