In General Relativity, which is our current description of gravity, changes in the gravity field is mediated by gravitational waves.
1. What are gravitational waves:
If you have any medium, and you add some energy to it, it starts to oscillate or wiggle. In other words, there is sort of a disturbance in the medium. This is true for air, water and even massless fields like electromagnetic fields. These disturbances are waves.
In GR, gravity is described as a property of the geometry of space and time itself. Any matter or energy can change the geometry of spacetime, and this is what other objects experience as the 'gravity' of the first one.
Spacetime is a medium in GR. It can ripple, it can be disturbed. Indeed, in situations where there is movement of energy, the spacetime around it can 'squish and squash', causing a wave of spacetime itself.
Notice, I did not say a wave in spacetime, because the spacetime itself is waving. Space and time expand and contract. That is a gravitational wave.
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What is the speed of gravitational waves:
If the spacetime itself is waving, then how do we define the speed of those waves?
Simple put, we take two objects separated by a long distance and interacting through gravity. Now, if we slightly change the position of the first one, the second one will not immediately change its motion. Instead, there will be a time gap. The distance between the two objects (before and after the change doesn't matter as we only slightly changed it; also the corresponding change in movement is small).
Now the large distance between the two objects divided by the small time gap gives us how fast gravity can propagate (through gravity waves), in other words, the speed of gravity,. That is how we define it: in terms of the distance between objects and the time gap between movement of a large object and corresponding effect on the smaller object.
3. Trivia: What does that come out to be?
As it turns out the speed of light $c$. This is because of the fact that waves can be thought to have 'mass', which is basically the mass of the medium (remember that the wave is nothing but a disturbance in the medium; so the mass of the wave is the mass of the medium, in a simplistic sense).
Now, electromagnetic fields have no mass and the wave in the fields is light. Special Relativity tells us that anything without mass travel at the speed of light, a constant $c$.
Space-time itself is massless too; it may contain mass in it, but that does not mean that it itself is massive too. Okay. So a wave in spacetime, is massless, and by the laws of physics, can only travel at the speed of light.
Ta-da! 343