Einstein, in his 1905 paper On the Electrodynamics of Moving Bodies, famously uses two postulates to derive the Lorentz transformations:
How are these postulates used to show that no signal may propagate faster than light?
Technically, they do not. Neither version of Relativity directly forbids FTL signalling, and are entirely consistent with so-called "tachyons," particles that travel faster than light.
However, FTL propogation interacts very badly with the Lorentz transformation, the mathematical relationship that translates coordinates in one frame to another while maintaining, the fixed speed of light. For background, Lorentz transforms often involve scaling by the "Lorentz factor",
$$ \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
From the Lorentz transform's properties, you can deduce that,
Any objects traveling at lightspeed will be seen to do so by everyone and cannot be accelerated in any way; conversely, objects starting out travelling slower than light cannot be accelerated to or beyond the speed of light because their momentum and inertia scale with the Lorentz factor as they approach light speed, and you find an infinite asymtote there as $v \to c$. Thus, if tachyons were to exist, they must have imaginary mass, and furthermore will slow down as they gain energy.
If an object were to take what one frame interprets as an FTL trajectory, i.e. moves forward in time but further in space than light would do in the same amount of time, then in a different frame, this appears as the object travelling backwards in time. Because this is a frame-dependent effect, an object could perform an FTL "warp", adjust its slower-than-light velocity to put itself in a carefully chosen different frame, and then "warp" in the reverse direction, and arrive at its origin before it left.
There is no current experimental evidence that tachyons exist, and the imaginary mass they would have to have causes problems if you try to fit into existing particle physics. (e.g. if we apply a real force to an imaginary mass, do we get an imaginary acceleration...?) Since known objects cannot be accelerated beyond lightspeed for the reasons mentioned, in practice FTL signalling is impossible even though it is consistent with the theory of Relativity. Additionally, the time travel entailed by FTL speeds makes physicists extremely skeptical of the idea as a matter of principle - it's important to the foundations of doing science that causes follow effects, after all.
