As a non-expert, I'll give it a try in simple language to provide something to work with, and gladly entertain any and all corrections...
All unaccelerated observers despite moving at various relative speeds among themselves always measure the speed of light* to be the same.. that is the starting observational thing to be accounted for. The first problem is that if you assume that time, distance, and simultaneity are absolute everywhere for all these observers then this does not make sense, seems paradoxical, and one must come up with some adjustments that would invent some very strange properties of light to account for it.
Relativity takes the observations as fundamental and the approach that the speed of light is assumed to be a constant as has always been observed... and then accounts for this by taking away the assumption of the absoluteness of time, distance, and simultaneity. This is revolutionary, but it ensures that all observations of light speed are constant and shifts the needed adjustments away from light and into the time, distance, and simultaneity calculations, with equations and concepts of time dilation, length contraction, and disagreement on simultaneity when one observer is accounting for what is happening to another.
Relativity adjusts these formerly absolute concepts in order for light to be assumed constant speed as observed. The nature of these adjustments conspire perfectly to make all light speed measures consistent for all observers, and amazingly the effects of these adjustments become transparent and unmeasurable to the one apparently subject to them when being measured and observed by another. This works so well that two observers will each observe and measure these adjustments in the other, but not in themselves.
This whole approach was based on a very careful step by step analysis of exactly what it means to measure time, distance, and simultaneity using light and the assumption that light speed is always constant.
* Light remains a peculiar thing in that one can't measure the one-way direction speed of light; the speed is always calculated from a two-way direction measurement (in which the speed in both directions is assumed to be the same).