When a rubber band is stretched out, there aren't as many ways the individual molecules can arrange themselves as there are when the rubber band is not stretched. They have to be sort of lined up, instead of in a jumble. There are a lot more ways to throw the molecules into a jumble than there are to line them up.
When there are more ways to arrange the molecules, the entropy is higher. So when you stretch a rubber band, entropy dictates that the rubber band will want to contract again. Where does temperature come in? When the temperature is higher, the molecules are more excited, and want even more to be in a random jumbled state. Think of a class of kindergardners on a sugar high--they'll want to run all over the place, not line up in a straight row. This makes the rubber band harder to stretch out, because it is harder to line up the molecules inside. Hence, it is stiffer at higher temperatures.
Things other than rubber bands can be described by the same entropic theories. They are even being used now in biology, to describe the behavior of DNA when stretched, for example.
There is a brief summary at this website.
Answered by: Danielle Cook, B.S., Bioengineering Graduate Student, MIT, Boston