A catalyst speeds up a reaction by increasing the number of successful particle collisions between reactive substances. In a typical reaction, the majority of particles do not have enough energy to react, and thus they simply bounce off each other. The catalyst provides an alternative reaction path with a lower activation energy, increasing the number of particles that can overcome that barrier and have a successful collision.
The University of California-Davis details a metaphorical explanation that can make this process easier to understand. Suppose there are two towns on either side of a large mountain, and the only way for people to get from one town to the other is by going over the mountain. Only the highest-energy people are able to overcome the barrier and get from one town to the other. Now, suppose a tunnel was cut through the mountain, providing an alternative path for the people to move from one town to the other. The tunnel is the catalyst because it provides a lower-energy route to allow more people to get from one side to the other. This process uses a different method to produce the desired result in less time, but without altering the original barrier.