I would not say PbtA are necessarily the "best designed" rpgs (best according to which criteria?), but they are certainly examples of successful game design.
The reason for their success is that the rules of the system supports effectively the design goals. Broadly stated, the main thing that the pbta system tries to do is creating a dramatically satisfying story whose protagonists are the PCs. The rule system works very tightly to promote this. Apocalypse World is one of the best systems to showcase this.
First of all, the rules manage to distribute narrative authority effectively without being too intrusive into the conversation. See, back in the day, some people had a problem with traditional games (like D&D or Vampires). They felt that the GM had too much power. In some of the older games, the rules gave explicit permission to the GM to do whatever he wanted, even changing the rules on-the-fly without informing anyone (the so-called Rule Zero). This was in the context of an approach to role playing games where players were expected to follow the pre-planned story of the GM - historically, this approach is generally traced back to the Dragonlance modules. While this might work perfectly fine for some people, other people had bad experiences with GMs abusing this rule. They lamented that the storytelling activity was essentially unilateral, and that non-GM players had no meaningful way of contributing to the story. So they started designing RPGs that challenged this framework. These RPGs manage to redistribute narrative authority among players, but they also have some pretty unconventional rules (see eg. Fiasco). PbtA manage to redistribute narrative authority effectively, at the same time retaining an almost traditional structure. You might actually not realise what they're doing until you think about it. How they do this? First of all, the GM can't do whatever he wants. The GM has to follow specific rules that dictates what he can say and when (Agenda, Principles, and Moves). This is very important. In AW, the rules say this (emphasis is mine):
There are a million ways to GM games; Apocalypse World calls for one way in particular. This chapter is it. Follow these as rules. The whole rest of the game is built upon this. Make Apocalypse World seem real. Make the players’ characters’ lives not boring. Play to find out what happens. Everything you say, you should do it to accomplish these three, and no other.
This is to be taken literally. Like, these are not advices or suggestions. These are, literally, rules. When you GM AW, everything you say must be aimed at literally accomplishing these three things. If you don't do this, you are cheating. See what I mean? The GM can't say or do whatever he wants - his powers are limited by the game.
Now, let's look at moves. This is probably my favourite move from the entire game. It is one of the moves from The Chopper. The Chopper is one of the playbooks (classes or archetypes), and is the leader of a gang. Here's the move:
Pack Alpha: When you try to impose your will on your gang, roll+hard. On a 10+, all 3. On a 7–9, choose 1: * They do what you want (otherwise, they refuse) * They don’t fight back over it (otherwise, they do fight back) * You don’t have to make an example of one of them (otherwise, you must) On a miss, someone in your gang makes a bid, idle or serious, to replace you for alpha.
There are a number of things that are really cool about moves from the design perspective. First of all, they tightly couple rules and narrative - you can't say that that you use "Pack Alpha", your character has to do something in the fiction of the game that triggers the move (i.e. you must describe what exactly is your character doing that counts as "impose your will on your gang" - asking nicely? that doesn't really count). The second thing is that they seamlessly redistribute narrative authority - when you do something in the fiction that triggers the move, the move is triggered. Based on the results, the player has a degree of narrative authority in that you have a (limited) saying in what happens as a consequence of something you did ("on a 7-9, choose 1"). The third thing that moves do is that they directly manipulate the fiction of what happens in the game. If you want to create a a dramatically satisfying story, you can easily see how a move like this will easily create drama and conflict. Not only that, but you can use moves to reinforce genre-specific tropes (the brutal leader that has to "make an example" of a gang of savages to impose his will is an example of these tropes).
These are just the main elements - if you look in depth at the pbta system you can find a lot of interesting features that make these games good examples of successful game design. You can understand why they became so popular.