Relativity explains a couple of counter intuitive things that we actually see:
1. Apparently there is no particular velocity that can be called absolutely stopped or special in any way: every velocity is just as valid as every other.
2. Observers moving at different velocities disagree on 3D distances and time intervals between events.
3. But they all agree on the spacetime distance between events. Which is to say, there is a simple formula that resembles a distance formula in 4D where one of the dimensions is time.[0] We know that using this formula, all observers at all velocities get the same result.
gravity is not being discussed but relativity addresses it as well.
So relativity explains actual (but counter intuitive) observations. If we had the simpler Newtonian universe then we could fly around at unlimited relative speeds and time would flow unambiguously at constant rate and be completely unconnected to distance/space. But by direct observation, we see we don't live in such a universe.
If causality could percolate at unlimited speeds then we would not observe any these things and the formulas for relativity would not work but the formulas of Newton would. For one, there would be infinite frames of reference for which you arrive before you depart and these frames of reference could use faster than light travel reverse casualty.
[0] https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval