Every chemical reaction needs a little "kick start" to begin. The energy it takes to get started is called the activation energy. The more energy the reactants have at the start, the more likely it is that they'll be able to get a reaction started. Temperature is a measure of energy, so it makes sense that reactions depend upon temperature.
Every chemical reaction starts with at least two separate reactants -- atoms or molecules -- and ends with one or more products. The reactants are in balance at the start, so the first step of a chemical reaction is to break up the existing balance so the reactants can be changed into a new configuration.
It's something like what you'd have to do to make a fruit salad from one can of diced peaches and one of diced pears. The cans are stable -- nothing will change until you put energy into them by opening them up. After you open them and pour them both into a bowl you create a new stable configuration -- nothing will change as the bowl sits there. But to change from the stable starting configuration to the stable ending configuration you needed to put energy in. That's the activation energy.
On a chemical level there's no one to add energy to the reactants, so they have to do it themselves. They do this by slamming into each other. The faster they slam into each other, the more energy in the collision. If a collision is fast enough to overcome the activation energy then the chemical reaction will take place. The link to temperature is that -- at its heart -- temperature is a measure of how rapidly molecules are moving.
As the temperature rises, the number of particles -- atoms or molecules -- with a certain amount of energy rises exponentially. That is, when you double the temperature, you don't double the number of reactants with enough energy, you increase the numbers much more quickly.
Reaction Rate with Temperature
You can see an extreme example in the chemical reaction of combustion. If the temperature in a room gets hot enough, in an instant all the flammable material will ignite. A less observable effect is the balance between polymerization and depolymerization. Polymerization is the chemical reaction that drives the assembly of big molecules composed of identical or very similar units. At a certain temperature -- different for each polymer -- the polymer will split into its components at the same rate at which it combines. Above that temperature the polymer will break up.
Enzymes are biological catalysts that decrease the activation energy of a chemical reaction. Essentially, enzymes shift the whole temperature curve downwards, so that a reaction at, say, 37 degrees Celsius with the enzyme happens at the same rate at which it would occur at, say, 140 degrees Celsius without the enzyme. Enzymes are large proteins that work because of their shape, and they lose their shape if the temperature gets too high. So an enzyme-mediated reaction will occur faster as the temperature increases, but then the rate slows down as the temperature gets higher. Above a certain temperature, the enzyme loses all function.