We all know that Einstein is a pretty smart folk, he is well known for his E = mc² formula and for his pictures
Einstein with his seducing legs
But why did he came up with relativity?
Ether
Around the end of the 19th century all the folks that were doing science thought they had all figured out. They thought that space was filled with a continuous medium called ether. Light and radio signals were all waves in this medium, just like sound is a pressure wave in air. All they needed to complete this theory was to mesure the elastic properties of the ether and boom, everything else would fit. Eh, it was not right.
Light always screw things up
Of course, one way or another, scientists would know they were wrong, but they started to notice something were wrong when they began doing experiments with light. In this theory we’d expect light to move at a fixed speed since the ether is continuos and if you were traveling to the same direction it would appear slower, just like we learned at high school. But not even one experiment managed to find evidence for differences in speed due to motion through the ether.
Electromagnetism
From Maxwell’s equations we can calculate the speed of light from some constants. But now you have a problem; what happens if you are in a moving vehicle at constant velocity in a straight line and you do electromagnetic experiments? If the speed of light changes in that reference frame it would also mean the constants of electromagnetism would change and therefore the whole theory of electromagnetism would break down (the equations would all become velocity dependent whereas the formulation of say, the electric field, isn’t normally). Basically, Maxwell’s equations are invariant under Lorentz transformations, but aren’t under Galileo transformations, which are invariant under the classical mechanics. So the physics world was torn, some equation had to be changed, or Maxwell’s or classical mechanics’. Einstein simply decided that electromagnetism is more fundamental in physics and then showed that classical mechanics could be modified in a way that it, too, became Lorentz invariant. In the end of the day we got relativity, electromagnetism was saved, classical mechanics were changed, the ether problem was fixed and Einstein got known for being a genius. And a photogenic physicist. 
Sources: StackExchange, Reddit, Wikipedia
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