The AEGIS interceptors are not as sophisticated as you'd think. All of the directing comes from the ship's powerful radar. The ship tracks the inbound missile, and when the timing is right launches an interceptor (the Navy calls them "Standard Missiles" or SMs). The missile has no idea where it's going or what it's supposed to hit, in fact it doesn't even know where IT is. It's only real link to the world is it's ability to listen to the ship's powerful radar. The same radar that detected the threat can also see the SM. It basically hits the SM with radar in a special way that tells the SM "go up", "go down", "turn this much" etc. and guides it into the target. At the very end of the flight there is a terminal phase that is a bit different, but it's still the radar doing the heavy lifting, not the missile.
Most of this is called out in more detail in the wikipedia page for Aegis Combat System.