It is accepted that the speed of light is the speed of causality. If we exceed the speed of light, the order of cause and effect breaks down. This happens as we see our surroundings moving backward in time. Right?
However, how do we know they move back in time once we move faster than light? If $v>c$, then in the formula $$ t'=\frac{t}{(1-v^2/c^2)^{1/2}} $$ we get an answer that is not defined, so how do we know time moves backwards?
The issue isn't whether time will move backwards at the speed of light, it's that having stuff sending signals around faster than light causes problems even for us nonrelativistic slower-than-light beings.
Consider a particle moving at speed $v$. It traces out a worldline $(t,v t)$ in some frame of reference, which I'll call frame 1. That is, at time "t" it will have a time coordinate of $t$ and a space coordinate of $vt$. You can also say that this traces out a "series of events", each event is at the coordinate $(t,vt)$. If it moves faster than light, there is no problem. As the time coordinate increases when $v>c$, we still are looking at a point more in the future. So there's no paradox yet and no movement backwards in time.
To another observer with speed $|u|<c$, however, this gets Lorentz transformed. To him, in frame 2, the coordinates appear at locations $(\gamma t-\gamma \frac{u v}{c^2} t,-\gamma u t+\gamma v t)$. $\gamma$ is well-defined, because here $\gamma=\frac{1}{\sqrt{1-u^2/c^2}}$. But the problem is that in this observer's frame, if $1-uv/c^2<0$, the particle moves backwards in time as $t$ increases! That is, in frame 1 the particle evolves forward in time, in frame 2 it evolves backwards in time.
If you follow this line of logic, you find that if you are allowed faster-than-light travel in arbitrary reference frames, you can cause paradoxes. eg, you could kill yourself before you can go back in time to kill yourself! I outline exactly how you can do that in this linked answer. That's why things like the Alcubierre drive are still safely in the realm of science fiction. Even though it's consistent with general relativity, if it was possible to create and destroy FTL drives arbitrarily, you would still get those paradoxes.
In response to the comment asking for more intuition. I can't give an intuitive "why" because special relativity totally breaks people's intuitions! But I can give an intuitive "how".
Say I have a special faster-than-light bomb. In my frame, causality makes sense: I launch it at a planet one light-year away, and in half a year (faster than light) the planet blows up. My description of the universe makes sense, because the planet blew up after I launched the bomb.
In your frame, if you're moving fast enough relative to me, you observe the planet explode, then the faster-than-light bomb travels back to my planet, then I press the button to launch it.
You can see how this causes problems, and yet no observer travels faster than light, so $\gamma$ is always real in any Lorentz transformations. The proper time of the bomb won't be real, but we don't need to take that into account.