What's the problem with light traveling at speed higher than $c$?

There were a couple of clues that Einstein and others found that led to the conclusion that the speed of light was special.

First, using Maxwell's equations, you can derive the existence of electromagnetic waves that travel at $$c = \frac{1}{\sqrt{\mu_o\epsilon_0}} \approx 300,000\, \textrm{km/sec}$$ where $\mu_0$ and $\epsilon_0$ are constants you can measure with simple electrical experiments. But, this is weird. All velocities are measured with respect to another object. If I'm traveling at 60 mph down the freeway, that 60 mph is with respect to the Earth. What is the speed of light relative to? Later experiments showed that it was neither the light emitter nor the light absorber. Rather, both measured the same speed of light regardless of their relative motion. This culminated in the most famous null result in physics: the Michelson-Morley interferometer experiment.

Second, Einstein imagined what he would see if he could catch up to a beam of light and fly along side it to see what it looked like. He realized that he would see a motionless wave of electric and magnetic fields. But, this is impossible according to Maxwell's equations. The equation that tells how electric fields are generated from charges $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0},$$ where $\rho$ is the local electric charge density, means that, in the absence of electrical charges, $$\nabla \cdot \vec{E} = 0$$ there can be no points in space where the electric field is maximized or minimized. So, Einstein reasoned, if a maximum field is impossible to observe, and this is what you would observe if you caught up to a light beam, then it must be impossible to catch up to a light beam.

Once a maximum speed is accepted as a basic fact about our universe, space, time, and how observers measure space and time must be rethought. This resulted in the Special and General Theories of Relativity.

One argument goes as follows:

Maxwell's equations predict many things, but you can massage them into a form exactly like an equation that describes a broad variety of waves. Call this "the wave equation".

So imagine you have a container of water and you generate ripples. You could move your head and travel along with one ripple, so that from your point of view the ripple appears stationary.

Now electrodynamics (light), and small waves, both obey the same equation. You can move your head along a ripple of water so that the ripple appears still. Can you move your head along a ripple of light so that the ripple appears still? Experimenters wanted to try this, and they found that no, you can't.

This experiment is now used to motivate the postulate that Maxwell's equations are the same in all frames of reference. This then implies that you have to use a Lorentz transformation instead of a Galilean one.

So what happens if you use special relativity to allow faster than light travel? This is answered in the question, "Can FTL-Communication between two points in the same frame of reference break causality?". If you only allow objects to travel FTL in a single privileged reference frame, you get no paradoxes. But if you allow FTL travel in arbitrary reference frames, you allow time travel and "killing yourself before you traveled back in time to kill yourself" paradoxes to boot!

And that's why the idea of FTL travel is so appalling to physicists.