In very rough terms, it's like being able to catch a ball - you watch it to see where it's going to go, and then put your hand there.

With a comet, measurements of its movement are taken, they plot out the path it takes, they determine how the gravitational effect of the Sun and planets will affect it, and then see if the estimated path intersects with where the Earth will be.

Yes, it's a bit complicated (computers help), and there are always things that can mess up the predictions, but that's the overview.