It is not so much a "logical explanation" as it is a hypothesis.
It was already known (with experiments with radio signals which are themselves photons of longer wavelength than visible light) that photons appeared to always travel at the same speed, regardless of the motion of the transmitter and the motion of the receiver. And this apparent speed was the same for all wavelengths used in the experiments.
This speed was called "the speed of light" and calculations showed that this speed should also apply to visible light, since it was the same kind of electromagnetic radiation.
When Einstein wrote his Theory of Relativity, he asked himself:
--what if the speed of light really were a speed limit in our universe, and
--what if that speed looked the same to all observers, regardless of their relative motion?
He then used mathematics to answer his own questions of "what if".
So far, his theory only described "what kind of universe would we have if the hypotheses turned out to be true".
Since then, scientists have been comparing the theoretical description with how things behave in the real universe... and they match (at least, as precisely as we can measure, up to now).
Many scientists are still looking for any sign that the theory is wrong... and have not found any yet.
The "speed of light in a vacuum" is the speed limit. We call this speed "c"
Light itself sometimes goes slower (for example, in water, light travels only at 3/4 of c).
Some particles come out of radioactive material at 0.99 c (99% the speed of light). Because these particles have some mass, it takes them some time to slow down if they meet with resistance.
Radioactive material put underwater will emit particle which (for a brief fraction of a second) will travel faster than the speed of light
0.99 c is faster than 0.75 c
look at:
https://en.wikipedia.org/wiki/Cherenkov_radiation
Light itself (in the form of radio waves) have been observed to travel faster than c (in old radar guide tubes, for example), but the information being carried does not travel faster than c. And that is OK under the theory. This was discovered during the Second World War. Einstein was still alive when the information was finally made public, and he had no problems with it. As long as the "information" carried by the photons did not go faster than c, the theory was OK.
In fact, the theory of Relativity does not specify "which way" the speed limit works. The theory does allow particles going faster than c (they would be called "tachyons"). The "speed limit" states that such particles could not slow down to below the speed of light. Therefore, the speed of light would remain a "speed limit" and tachyons would be unable to slow down past this limit.
But tachyons have never been observed.
So, the mathematics are not an explanation. There are only a way to apply the concept of a speed limit, to how things should behave if that speed limit really existed in our universe.
Because the universe seems to behave in the same way as described in the mathematics, we conclude that the idea of a speed limit could be true.