You can think of the speed of light as the speed at which causes produce effects. In the same way that the speed of sound[1] is the speed at which one air molecule can tell another air molecule that it's been been moved, the speed of light is the speed at which one piece of spacetime can tell another piece of spacetime that something has happened to it. Light just happens to move as fast as it is possible to move, the speed of light might be better described as the speed of causality.
This speed defines the "happens before" relation in spacetime, like a Lamport clock defines the "happens before" relation in a distributed system. Event A "happens before" event B if all the information about B was known when A happened. Otherwise B happens at the same time or later than A.
If you can travel faster than light (faster than causality), there's nothing stopping you from constructing a scenario where A happens before B and B happens before A. And that's what it means to violate causality.
[1] To head off a potential confusion, note that while a jet may be able to move faster than sound, the sound of that jet passing by does not.
To an observer who is only able to sense sound, in his reality nothing should be able to move faster than the speed of sound because otherwise he'd observe event B happening before event A. Of course in our reality moving faster than the speed of sound is possible. And in terms of sound we hear the object actually moving back in time. That is we hear it first when it's at our location and then we hear the sound it made further and further back in time. From the jets point of view though he was not moving backwards in sound-time.
For us, the same would apply to something moving faster than the speed of light. We would observe it moving backwards in time but that does not mean it actually does from its point of view.
Point being we have to be very careful about distinguishing between when we observe something happening and when it actually happens. Observing something going backwards in time is fine while actually traveling backwards in time probably is not.
I think you can observe this effect in the high speed Schlieren footage of a whip in this video https://youtube.com/watch?v=AnaASTBn_K4 (you can skip to around 8 minutes) but I'm not sure, I keep going back and forth on it.
You cannot hear any sound as it approaches. When it is at the cameras location, you hear a boom as a lot of soundwaves hit you at nearly the same time. After that you hear a mix of the soundwaves that the jet produces while traveling away and the other part is the soundwaves arriving from earlier during the approach. Unfortunately we can't really distinguish between those with our ears though a machine could. With light you would see indeed the object suddenly appearing out of nowhere and then splitting into two. One traveling into the direction of time and one seemingly going backwards.
What makes things a bit weird are for example time dilation and length contraction. Our equations for those would result in imaginary numbers (e.g. with "i") and at least to me it's a bit unclear as to what that would mean.
It is also unclear to me if a particle traveling faster than light would still see light traveling at the speed of light from its frame of reference.
From the perspective of a being that can sense only sound the hypersonic traveling jet would be equally confusing - it could assume that there must be things happening in another dimension, one that it cannot sense/observe. And so one could theorize that an object being able to travel faster than light could exist in a dimension outside our 4 spacetime ones - one that we cannot observe directly.
This speed defines the "happens before" relation in spacetime, like a Lamport clock defines the "happens before" relation in a distributed system. Event A "happens before" event B if all the information about B was known when A happened. Otherwise B happens at the same time or later than A.
If you can travel faster than light (faster than causality), there's nothing stopping you from constructing a scenario where A happens before B and B happens before A. And that's what it means to violate causality.
[1] To head off a potential confusion, note that while a jet may be able to move faster than sound, the sound of that jet passing by does not.