Running puts a lot of stress on your joints, and taking rest days will give your joints a chance to recover from all that pounding. It's also good to take a mental break from running, so you don't lose motivation by running every day.
Most runners need at least one, even two, days off from running and other exercise. Research has shown that taking at least one day off a week reduces the frequency of overuse injuries, such as shin splints and stress fractures. Even the most elite runners take rest days, although rest days for them usually involve a low-impact cross-training activity, such as swimming. Those types of cross-training days are considered to be rest days because they give the joints and muscles you use in running a break.