How does an increase in temperature affect the rate of a reaction?

Increasing the temperature of a reaction means increasing the kinetic energy of each particle in the reaction. Increasing a particle's kinetic energy means the particle is moving faster ('Kinetic' indicates motion). So increasing the temperature a reaction is performed at increases how fast each particle involved is moving, this increases the rate at which collisions occur (a larger number of collisions results in a larger number of successful collisions) and increases the energy of the particles in each collision (more energetic particles results in a larger fraction of collisions actually being successful). Both of these factors work to increase the rate of reaction, so increasing the temperature a reaction is performed at increases the rate of the reaction.
An analogy I like for this concept is Dodgems/bumper cars. If all the drivers in a bumper car arena start driving faster, they are far more likely to collide with each other – not only that, but their collisions will also be a lot more energetic. I enjoy using this analogy because it does a good job linking the abstract with something fun that most people can imagine clearly. It can also be extended to the other ways the rate of a reaction can be changed (i.e. increasing the pressure is analogous to decreasing the size of the arena while keeping the number of cars constant and increasing reactant concentration is analogous to increasing the number of cars while keeping the arena the same size).