If nothing travels at the speed of light, except light, how can a black hole also pull light into itself?

Gravity is a force, and it need not have a "speed"¹.

A gravitational body sets up a gravitational field around itself. Note that by the time the light wave approaches the body, this field has already been set up. The gravitational force does not need to "reach out" and "catch up to the light" — it's already there.

In fact, it's not even really a force (though we can treat it as one to an approximation). Gravity bends the fabric of spacetime around it, messing with the meaning of a "straight line". From different reference frames, different lines appear straight. To the light wave, the inspiraling path seems "straight" and thus it follows that path. To an external observer, the light wave is not going straight.

^{1. It does, but that's the speed at which changes to the field propagate, not the speed at which it "catches up" with other things. Changes in the gravitational field propagate at lightspeed.}