Everyone has covered image clarity, aperture, and F-Stop, but there's three more advantages as well:
A basic prime lens (Canon 50mm 1.8) costs $99 and is a fantastic value. You can buy a cheap prime to experiment with to see if you like prime lenses, before you commit to getting higher-quality prime lenses. There are no zoom lenses cheap enough to throw away, you'll have to rent.
Primes can fill out your kit until you commit to bigger budget. I love the Canon 24-70 f/2.8 L II lens, but you shouldn't buy that expensive a lens until you find you need it often. Instead, using a 35mm prime lens and a 50mm prime lens can cover your indoor, low-light shooting.
Physically, prime lenses are short and light. You can add several to your kit without having to buy a bigger bag or get into better shape.
Prime lenses are more efficient with light than zoom lenses, so in the same amount of light, you can use a wider aperture (lower F-stop), and get a blurrier background. Or a faster shutter speed, and get less motion blur. Or a lower ASA, and get fewer image noise artifacts.
I love prime lenses! When you get a few primes, you should practice switching lenses smoothly. I recommend this article: http://strobist.blogspot.com/2006/06/pro-tip-how-to-quick-change-slr-lens.html