The beginning is the best line, that no one really knows "why" and what I have said to even other physicists who don't really understand it. A lot of the counterintuitive consequences of the paradox of the moving charges detailed here are a fact of nature but we don't really know why it must be that way. For example, in a modern particle physics starting point for qed, we force the equations to be lorentz invariant from the start but that is a starting assumption. But that's it, there's no real deeper answer to "why" beyond it being a experimentally observable fact, and the only resolution of these observations is that lorentz transformation and thus things like time dilation and length contraction must happen.
Small relativity related tidbit: I hate when people say the phrase "the faster you go, time slows down for you." This is a problem because it implies that the moving observer notices their own time dilation which is reverse of the case. Of course, every non-accelerating observer is in their own rest frame, so it doesn't make sense to say "time slows down for you," because you are your own reference and there is no other frame to base your measurements on (I mean that was the whole point of relativity, there is no universal rest frame). Instead, when you measure the rate of change for other reference frames moving relative to you, their clocks move slower when measured by your clock. So the actual phrase should be something like "the faster others go, the slower their time appears to you."
Small relativity related tidbit: I hate when people say the phrase "the faster you go, time slows down for you." This is a problem because it implies that the moving observer notices their own time dilation which is reverse of the case. Of course, every non-accelerating observer is in their own rest frame, so it doesn't make sense to say "time slows down for you," because you are your own reference and there is no other frame to base your measurements on (I mean that was the whole point of relativity, there is no universal rest frame). Instead, when you measure the rate of change for other reference frames moving relative to you, their clocks move slower when measured by your clock. So the actual phrase should be something like "the faster others go, the slower their time appears to you."