Our daily experience is quite adequately explained by Galilean relativity and Newtonian physics, without any need to refer to Einsteinian relativity.
In 1845, Urbain Leverrier observed a 35" per century excess precession of Mercury's orbit, a discrepancy that could not be explained until Einstein did so.
In 1882, Simon Newcomb observed a 43" per century excess precession of Mercury's orbit, a discrepancy which lacked an explanation in normal physics.
At the start of the 1900s, it was clear that there were a number of unsolved problems which needed a new theory or theories to explain them adequately.
The first problem was that the speed of light did not behave like the speeds of ordinary objects made of matter, because it seemed to be constant.
The next problem was that some forms of matter, in particular the radioactive elements, showed a powerful and seemingly unpredictable instability.
The next problem was that there was no way to account for the way atoms emitted light and other forms of radiation when they were heated or excited.
The next problem was that Newton's laws, which generally worked perfectly well, could not explain oddities in the precession of Mercury's orbit.
The last problem was that the heat capacity of molecular gases was not what it should have been, using calculations based on Newtonian theory.
In 1907, Albert Einstein offered his equivalence principle (gravitation and inertia) and predicted the existence of gravitational redshift from this.
In 1915, Albert Einstein completed his theory of general relativity, predicted light bending and offered an explanation for perihelion shift of Mercury.
Einstein's theory of special relativity indicated that time was relative, that the speed of light was constant, and that mass and energy were equivalent.
Physicists say if Albert Einstein had not proposed special relativity, the time was right and Hendrik Lorentz or Jean Perrin would have done so soon after.
Einstein's general theory of relativity offered the disturbing view that gravitation, rather than being a force, is a curved field in the space-time continuum.
The key effect of the two theories of relativity was to make us aware that space and time are not separate: they are an intertwined space-time.
In 1861 and 1865, James Clerk Maxwell showed from theory alone, that there should be electromagnetic radiation, and that light was part of it.
James Clerk Maxwell showed that his mathematically derived electromagnetic radiation would always travel at what we now call the speed of light.
If we see a ruler going past us at close to the speed of light, or if we pass one at that sort of speed, it will appear to be considerably shorter than it is.
The shortening effect comes about because the ruler's length relates to the average separation of the atoms, and in the direction of travel, this is shorter.
A person standing beside (or travelling with) the apparently shortened ruler, will see it at its normal length as they are in the same frame of reference.