I think once you understand the proof of 5 color theorem and believe the fact that the proof of 4 color theorem is more or less the same, but more hardcore, then it is as much insight as you can get.

Looking for insight is what many people do in mathematics, but nonetheless believing that every true fact is true for a reason is naive, just as naive as believing that every problem has to have a solution. So far, most of the time we succeed in explaining why true things are true, but it may be the case that some facts are true merely by combinatorics, and not for human-imaginable reason.

Sure, I would like to believe that there is a crucial observation we haven't made yet that makes 4 color theorem easy, just as it is for 6 color theorem. Nevertheless, I accept the fact that there may be no such thing and 4 color theorem holds just because the constraints force it to be.