For one thing, it perfectly captures the feel of the genre. The PCs are basically trying to identify and pick-apart a giant conspiracy, and this provides some truly excellent immersion into that mindset. Having used this now in multiple campaigns, I can't imagine running a "detective genre" game ever again without using it.
Along with that oh-so-tasty immersion in the setting, it also helps the players visualize what they are up against. The existence of this player-built map forces the PCs to have an ongoing dialog about the clues they've gathered and the avenues of investigation still open to them. It's a mystery game, so you want the players to take it seriously and try to figure out the big picture. The Adversary Map provides a guiding structure for those efforts, and it's damn awesome that the game rewards that good behavior in play.
I tend to enjoy running complicated and mysterious plotlines, and fully-fleshed sand-box settings with lots of nuance and considerable PC freedom. When a GM does that, though, you always run the risk that players will feel overwhelmed, or get mired in the detail. The Adversary Map generates a big-picture view so you've always got some common ground to start from. It also keeps people from focusing in on one small subset of the plot or setting to the exclusion of all else, because they get a subtle weekly reminder that there's a lot more out there on the edges that they haven't explored yet. Should the players get stuck anyway, it's an easy thing for the GM to point at two or three different pins or photos and say "if you start shaking the clue tree in one of these spots, I guarantee you something interesting will fall out". That's a little heavy handed, so I wouldn't do it unless they'd really hit a wall, but it's nice to know that you've got that option hanging on the wall to refocus them if the PCs are feeling lost or caught up in analysis paralysis. I love this this mechanic. It's both an immersive tool and a safety net for when I get too clever for my own (or my players) good.