The main idea behind weight classes is, of course, setting a level playing field for the different fighters. Another result of the weight classes is for fans to know how much the fighters weigh, making the sport a little more understandable and relatable. If I know that a fighter weighs approximately the same as me, I can compare myself to him.
Boxing, where weigh-ins started, was never a very institutionalized sport. It was always a sport in which fights were made ad-hoc, and in which the different promoters had to go through a lot of negotiations to be able to make these fights. There was never a very orderly, rank driven kind of management for matchmaking. So the treatment the weigh-ins always got was similarly not very professional. Fighters could do pretty much whatever they wanted, as long as they stand on that scale and make weight once, usually the day before the fight.
MMA adopted this amateurish attitude, because for a long time, MMA was also a sport without some central organization which could organize it all in a more professional, safe fashion. Now finally (and after ONE championship has declared its implementation of similar measures a few months ago) the UFC has announced new weigh-in rules, for the safety of the fighters.
I agree that the safety of the fighters is the most important thing. but there is another factor to consider, and that is the the effect on the fans.
