Great post above me by Army Eye, in all aspects.
As for my own take.. I have mixed feelings about the system. I do agree that the actual champion should be decided in a match between the current WC and a challenger, to me this is a no brainer. It is a great tradition and I don't see one reason to get rid of it.. However for the candidates cycle.. I am not sure about the candidate matches to be honest.
To me, the problem is that during the chess "season" (any one given year) there are literally no matches played, well despite some absolutely random promotional thingies like Karpov vs Hou Yifan right now. All top players do the entire year is play round-robin tournaments. Now any competent player would tell you that match play and tournament play are two very different animals. So how come after playing tournaments all year, suddenly to select the candidates they want to use matches?
I mean lets take a simple example: Let's say Carlsen absolutely dominates the tournaments for the entire year (something he has done, maybe not absolutely, but he did dominate), gets to No1 in the rankings and most people agree that he is the best player around ATM, while for example Kramnik has very average success, placing around the middle in most of those tournaments. Now the time comes, and Carlsen has to face Kramnik in a candidate match. We all know that Kramnik is a great match player and I would not be surprised one bit if he beat Carlsen like 4.5 - 3.5 or so. But how can we say that Kramnik is a better player, after Carlsen has been dominating the tournament circle for the entire year? For this one match, maybe Kramnik was better prepared, maybe he was a bit more in form, maybe Kramnik is just a bad opponent for Carlsen... Now the definition of "champion" should be "the best out of everyone/all". But to Carlsen, in this case, it is "the best against Kramnik!" as he does not even have a chance to prove himself against the others.
Now on the other hand, if you have a round-robin (or even double-round-robin) tournament, such things do not matter - one has a chance to prove himself against everyone and one such "miss-match" will not hurt your chances so bad. Plus, it seems a quite logical continuation for the season, when there are only tournaments and no matches.
Some people said that matches are more attractive to the spectators, but I don't necessarily agree. I don't know about you, but to me, the San Luis 2005 was one of the most interesting tournaments of all time. Those little words "world championship" give it a certain aura and makes it completely different from Linares, Bilbao, or any other tournament, at least to me. Also, a lot of matches among top GM's are quite boring affairs with a bunch of draws because there is no need to win too many games, all you need is to win one more than your opponent, while in a tournament were only a couple of first places advance there is a need to score wins, thus the fighting spirit is much much greater. Plus, as I said before, tournament eliminates certain miss-matches and lets us see who is the best out of everyone, not against a certain opponent.
Well this has turned in to TL;DR , but hopefully I made my stance clear