Hamilton, when healthy, is among the best pure hitters in the game, and if he's limited to left field in Anaheim will probably be worth about 5 wins a year above replacement, and roughly that much to the Angels assuming he takes away at bats from a mix of Peter Bourjos, Mark Trumbo, and Vernon Wells. (The outfield situation, I should note, could certainly change.) Hamilton did benefit from a ballpark very friendly to lefty power hitters in Texas but should still be good for 30 homers a year in the power-depressing park in Anaheim, with strong OBPs and plus defense if he's in left. He doesn't address the Angels' biggest need, starting pitching, but he was the best free agent left on the market and does at least free up some other assets, especially Bourjos, for the team to trade for another starter. Bourjos' main value is his centerfield defense, but with the AL's lowercase most valuable player in center, the Angels have no need for what Bourjos provides.
The problems with the Hamilton contract are fairly obvious -- he's not durable, and he's already at an age where previously durable players tend to become less so. Over the course of a five-year deal, the over/under on Hamilton's games played is probably 600, if not less, given his propensity for injury. The Angels using him in left field might reduce that risk a little given the greater range required of a centerfielder, but much of Hamilton's problem staying healthy is a function of genetics and his style of play. He also finished the 2012 season like one of the walking dead, apparently the effects of his attempt to quit using tobacco products. That was an admirable goal but the late-season malaise likely cost the Rangers the division title. Hamilton's deal also takes him into the age range where most hitters begin to decline offensively and often find themselves heading for the cliff, or over it; between that and his history of injuries it's hard to see him producing $25 million of value in years 4 or 5 even given the existing rate of salary inflation in baseball. The 2016 Angels are going to be paying a lot of money for three players who probably won't be very good given their ages at that point.
The signing makes more sense for the Angels if they now look to deal some of their position-player surplus, especially Bourjos, but also perhaps Mark Trumbo and the buried catching prospect Hank Conger. The Mets might be inclined to deal R.A. Dickey for a package involving some of those players if they can't agree on an extension with Dickey, and getting, say, Bourjos, Conger, and another prospect (I'll throw Taylor Lindsey's name out there) for one year of Dickey's services would make a lot of sense for a team that is still rebuilding.
Hamilton's signing leaves one everyday centerfielder, Michael Bourn, remaining on the market (assuming some teams saw Hamilton in that light), but few teams looking to spend on a full timer at the position. The Rangers would be one, having lost Hamilton and looking at internal options like Leonys Martin and Craig Gentry. The Reds have three everyday corner guys but no true CF on the roster, but would have to change course from their stated plans for center. Bourn might have to look for a one-year deal that puts him back on the market next winter, but that market also includes Jacoby Ellsbury and Curtis Granderson if neither signs an extension before then, putting Bourn in a difficult spot at the moment. One other team that could benefit from Bourn if he'll take a one year deal is the Royals, whose overhauled rotation would look even better with one of the game's premier flycatchers out in center.