There is an experimental fact: motion is not absolute, but relative. It's not possible to say if some object is moving or not. It may be moving relative to some other object and be in rest relative to some other object.
This is not just some obvious blah-blah-blah. But actually this is an amazing and counter-intuitive fact. For hundreds of years people were sure that there is an "absolute rest": if you stop pushing a wagon it will eventually stop and wouldn't move until you push it again. This is the case for almost everything you can see around. All the objects which nobody touches are in rest, and this is the absolute rest. That turned out to be wrong, all these objects are actually moving with the same speed as Earth (whatever this speed is).
One can argue: "Well, of course motion is relative, you can change position only relative to something, otherwise you wouldn't notice you position has changed. Nothing interesting." Again, this is not the case. If one is moving with acceleration he can understand it even not being able to compare it's position to some other objects. (General Relativity Theory says that things are little bit more complicated, but let's forget about it now). Acceleration is absolute. Sitting in a closed box you can tell if the box is accelerating or not and can find out it's acceleration, but you can't find it's speed.
Suppose there is a "truly resting" frame of reference. Being inside the box, how would you try to find out if you are moving or not? Well, by doing some experiments. Something like "I'll bounce these two objects, I know the laws of physics, written in the "truly resting" frame of reference, I can calculate how these objects would behave according to these laws, if I observe anything different that would mean my box is moving".
If you know laws of physics in some frame of reference you can switch the frame of reference and write the same laws using coordinates in a new frame of reference. You just have to know the transition rules. People used to think that transition rules are Galilean transformation: just replace $x -> x + v*t$ in the mathematical formulas of the physics laws and you get the formulas for new frame of reference.
It immediately turned out that the classical mechanics laws would not change when switching to another frame of reference moving with a constant speed. That means you wouldn't be able to find any differences between your predictions and actual behavior of the bodies and wouldn't be able to find out if box is moving or not.
Then people studied electromagnetism, found new laws of physics (Maxwell's equations) and it turned out that in moving frame of reference these laws have different form.
From "mathematical" point of view you substitute $x -> x + v*t$ in the Maxwell's laws and the resulting formulas would be different. More intuitively: Maxwell's laws predicts that there are electromagnetic waves which propagate with some constant speed. Same laws after substitution $x -> x + v*t$ should predict that speed of electromagnetic waves can be higher or lower by up to $v$ depending on direction.
That seemed cool. That would mean that there is a "truly resting" frame of reference: the one where speed of waves is constant. And even being inside the box you can find out if it is moving or not: just measure the speed of electromagnetic waves in different directions.
But the problems was that all the attempts to make such an experiment failed.
That seemed like a paradox. And the rootcause of the paradox was the obvious and well known transition rule we used to switch to a moving frame of reference: $x -> x + v*t$. That rules are not correct in out Universe. New rules were discovered, and that's how the Special Theory of Relativity was born.