Well sure, but is there some relationship between gravity and light other than that?
Sort of. There is no direct relationship between the two. In fact there's nothing special about light. Although we call c the speed of light, that's historical: we first connected c to the speed of propagation of electromagnetic radiation. But in relativity, c has a much more fundamental importance that has nothing to do with light.
This gets complicated, and I can't be fully accurate without bringing math into it, but the basic idea is that in relativity space-time obeys certain mathematical properties. One simplified way to think about it is to think about what "space-time" is. It is really a way to consider space and time to be fundamentally the same "thing" even though we ordinarily experience them differently. Most people have heard about ways in which under extreme conditions space and time start to change in odd ways, like at high speeds things shrink and clocks run slower. What you're seeing is a form of "exchange" between space and time: space gets "smaller" (things shrink) and time gets "longer" (clocks run slower). How you experience space and time depend on how fast you're moving relative to other things.
The "exchange rate" between space and time is c. In other words, in relativity 300,000 kilometers of distance is "equal to" one second of time. When you see "300,000 km/sec" you think "speed" but in relativity that's actually an exchange rate, like 1.13 dollars to the euro.
All events that spend less than 300,000 kilometers for each second are within the
causal light cone of that event. Colloquially if light can get there from here, we can affect that event. If not, we cannot. c, the "speed of light" is really the speed of causality, the maximum exchange rate for one event to cause or effect (affect) the other.
Electromagnetic radiation - i.e. light - travels at this speed. Gravitational waves, for separate albeit related mathematical reasons - also travel at this maximum speed of causality.
There's also special relativity which states that massless particles must move at exactly the speed c. Photons are massless and thus move at the speed c. Gravitons are theorized to be massless, and thus must also move at speed c. Why? More crazy math. But there are a few colloquial ways to describe the math. One is to say that it actually isn't the case that massless particles move at velocity c but the reverse: all particles that move at velocity c must be massless. You could use the lorentz equations to prove that any particle with any intrinsic mass that moved at that speed would then have infinite mass, which suggests a contradiction: that means the universe doesn't allow that to happen. And you can go backwards and say that if a massless particle is moving at the speed of light, it is impossible for it to ever change velocity. A massless particle moving at velocity c experiences no time due to lorentz effects: that's another way of saying its internal state can never change. A massless particle can hit something, vanish, and then be re-emitted. Its speed can *appear* to be slowed down because of these interactions. But really all that is going on is that it is created, moves at the speed c, then is destroyed having always moved at that velocity.
When you get deep into the physics, you start to learn that what non-physicists and even physicists colloquially call "the speed of light" actually is not about light at all: it is a fundamental property of the universe, that light just happens to honor. Light isn't special. c, the relationship between distance and time that happens to be about 300,000 km/sec in metric units, is special. It represents something more fundamental than a mere velocity. It is built into the fundamental nature of how everything in the universe can, or cannot, affect everything else.