Everyone has covered image clarity, aperture, and F-Stop, but there's three more advantages as well:
Price
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.
Price Again
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.
Size
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.
Bokeh
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