Most things in nature behave in a fairly predictable way. Most people can make predictions about temperature and how it will affect things: Heat expands, cold contracts. Watch a cake in an oven, and you'll notice that it expands as the batter becomes warmer. When many people consider a rubber band, they expect it to behave similarly, but one of the strange things about rubber is that it does the opposite.
The assumption is that a heated rubber band will expand. However, if you heat a rubber band with a hair dryer, you'll notice it doing exactly the opposite--it contracts! The rubber band will actually shrink because of the molecular properties of rubber. Rubber bands also become more brittle when hot--something you might associate with cold because you've seen ice shatter when hit by something solid.
Everything in the world is made up of molecules, and rubber bands are no exception. Molecules are tiny particles, made up of several different atoms. In solids, such as rubber, they are trying to fit together in one set shape. Millions and millions of rubber molecules arranged in a strip and joined at both ends form a rubber band. Molecules, however, are affected by temperature--they move faster with more heat, slower with less.
Under normal temperatures, the reason a rubber band snaps back is also due to its molecular properties. The strands that make up the rubber band are stretched, but they want to return to their regular shape. It's kind of like when you stretch after sitting down for a long time: Your arms are raised in the air, but you don't want to walk around like that (at least, not very often). Your muscles are more comfortable when in their resting position; the same applies to molecules in a rubber band.
Effects of Heat
The behaviour of molecules under heat would suggest that the rubber band becomes more flexible and wider. However, because of the arrangement of the particles, the band becomes smaller. It's kind of like a fence on a hill, when all of the nails are in place, the fence is sturdy, but if you remove those nails and make the slats more movable, it will eventually collapse down the hill--and take up less space. This is an imperfect analogy, as the rubber band will retain its shape, but the concept is close.
Effects of Cold
The converse is also true. If you cool down a rubber band, it actually becomes stretchier and expands slightly. This is because the molecules actually become more organised into a more efficient stretching shape. Because the molecules are more rigid and structured, they are actually stronger. Returning to the fence analogy, a straighter and better-constructed fence will actually be far more sturdy than a shoddily built one. Again, the analogy is imperfect, because fences don't stretch very well.