Staggered advancement is Pathfinder Unchained's answer to several of the other systems out there that have implemented systems that involve improving a character's abilities more often than at every level. Instead of gaining a whole swath of abilities upon advancement, PCs pick up abilities at 25%, 50%, and 75% of the experience to rise in level. In fact, the bulk of the system's 2 pages are explained in a giant table on pp. 42-43.
How It Works
Our friends at Paizo have broken the benefits you get into a few different baskets. Easiest are normal class abilities. PCs pick up their new class abilities when they level up traditionally. There's no changes here.
Skills: You pick up half of your skills at halfway to the next level and the other half when you actually level. Like everything else in RPGs, round down at the halfway point. So, if you are supposed to pick up seven skill ranks a level, you get 3 at 50% of the way there and the other four when you fully advance.
BAB, Saves, and hit points: These three are more complicated. First, you have to figure out how many of them advance per level (note that using fractional bonuses would alleviate this problem). If only one of them advances, you get it at 75% of the way there. If two advance, you get them at 50% of the way to the next level and at 75%. You only use 25%, 50%, and 75% if all three are going to go up. This is one of my few quibbles with the system. If you're only raising two, I'd raise them at 25% and 75%. After all, you're getting skills at 50% anyway. You advance the full BAB and the full saves, but get only 50% of hp (rounding down like always). You get the other half of your extra hp when you fully level.
Comparisons: Numenera and 13th Age
There are a couple of other systems that use staggered advancement that make for good advancement. I also think that they use staggered advancement far more effectively. 13th Age is the easiest because it doesn't use experience at all. Instead, at pre-determined moments the GM lets characters know that they've advanced a step. At that time, you check one of the boxes:
That's it. You check a box each time. When all the boxes are checked, you advance to the next level. You have to check all of them before you can move on to checking them again. So, if you start with a feat, you've got to check all the necessary boxes before you advance to the next level and take another feat. The game suggests that GMs might want to let characters level all at once sometimes--I highly suggest this for going from 1st to 2nd level.
In Numenera, the Strange and (presumably) the Cypher System writ large, players gain XP by accepting GM intrusions. The GM decides something interesting (bad) is going to happen and awards the player 2 XP. The player keeps one, but gives the other to another player at the table. Players can also earn XP at the end of sessions or at important story points. Players can spend an XP to reroll any roll. PCs can also spend 4 XP to gain a benefit: 4 points to allocate amongst your stats, adding one point to one of your character's edge pools, increasing your character's effort score by one, or become trained in a new skill. Numenera assumes that characters will spend 4XP to do this every level, but it might be fun to raise that by one point a level so that you need 5XP to pick up a benefit at 2nd level, 6XP at 3rd, etc. This also slows down progression because characters are going to be spending XP on rerolls like drunken sailors anyway!
How Does Pathfinder Unchained's System Stack Up?
In the end, I'm pretty meh on this system. It just doesn't make much sense in the Pathfinder universe. Staggered advancement makes a lot more sense in more story-driven systems. Pathfinder is a statistic-driven system though. The system feels arbitrary and it's too complicated: the bonuses come at different times and yet they feel too tightly prescribed at the same time. I wouldn't mind it if my players wanted to use it. That said, as a player I wouldn't want to!