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.

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.