One important engineering advantage is that they prevent a rocket motor from igniting explosively, and exploding. This is called a "hard start" and is one of the most destructive things that a rocket can do. However, hypergolic fuels are highly toxic and sometimes unstable, so a very high quality igniter, like a small rocket engine, is often used as an alternative.
Another important advantage is that the igniter cannot fail, so hypergolic fuels are the solution when the rocket must really start.
Some common hypergolic fuel combinations are: