Open world games are probably the most popular single-player experiences in the industry today. Many gamers, developers, and critics have begun to consider open worlds objectively superior to restricted ones. Of course, as something becomes popular, the laws of nature require a vocal minority to rise up and attempt to summarily discredit it, so we have traditionalists who lament the supposed aimlessness of these GTA clones. As usual, there's a ton of middle ground here - an open worlds isn't inherently good or bad, but a tool that must be used properly to add worth. So let's have a think about how, historically, these open world games drive players' interest.
An obvious pitfall in creating an open world is that they have a tendency to be too open - the player may lose interest if they aren't being challenged to progress and that progression doesn't have repercussions. One solution to this is what I would call the toy (or sandbox) approach - if it's fun to do everything, the player is guaranteed to be having fun, right? And if they're having fun, they'll keep playing. This is the modus operandi of series like Grand Theft Auto and Saints Row. While the world architecture is mundane, lifeless, and irrelevant to gameplay, the player can still have a great sandbox experience crashing cars, flying helicopters, and firing off rocket lawn chairs. The toy approach doesn't even really benefit from an open world - in fact, I'd posit that the open worlds are often merely an excuse to generate loads of content for an otherwise flat and underdeveloped mechanical core.
Needless to say, toys get boring, especially when one game after another is set in the same damn East-Coast-style city in the same damn contemporary timeline with the same damn cars and guns to play with. More ambitious (and traditionally rooted) games like Arkham City, Prototype, and Red Faction: Guerilla encourage the player to interact with the world, bestowing the environment with its own character. This can mean the traps and perches of Arkham, the shifting zombie/military dynamics of Prototype City, or the full-on destructibility of Guerilla's Mars. This is more than just adding an extra variable into the mix. Defining a unique and nuanced world that complements core mechanics is the very essence of open world gameplay - it's what makes freedom rewarding.
A great deal of the appeal of an open world game is scope. I'm not talking about the size of the world, but the size of the action. Even in missions, the player is free to choose their own turf*. This gives them the opportunity to make use of their knowledge of the world to alter the circumstances presented them. Take an example from Assassin's Creed; a mission might task the player with escaping a cadre of guards. A novice player may choose the simplest solution, to stand and fight. A player who better knows the world could make a run for a great hideout or find allies to help with the fight. The player is rewarded for capitalizing on their freedom and knowledge of the world and is thus encouraged to continue learning about it.
So one way of looking at it is that an open world game is driven by the urge to learn the world and use that knowledge to one's advantage. How much there is to learn and how well that knowledge can be used actually provides a pretty good metric of quality for an open world. In fact, it's really what differentiates an open world from a plain old over-world. Arkham City is loaded with tricks, traps, and gadgets that provide all kinds of ways to approach each conflict, rewarding the player for recognizing their surrounding and the tools it presents. At the other end of the spectrum, Sleeping Dogs' extremely rudimentary open world is really just an over-world. The player spends most of the time driving, which isn't even the main gameplay mode, and there's no way to take advantage of freedom-of-setting to win brawls or shootouts. Even the races are extremely tightly confined. Thus there's no reward for learning about the world - there's not even really any exploration, since you have an all-inclusive map and a GPS to highlight the quickest path. That's why Sleeping Dogs plays more like a traditional brawler/shooter equipped with an over-world.
The open world itself isn't really a style of gameplay (it doesn't fit the goals and oppositions model and can be characterized by some combination of exploration + another type of gameplay (shooter, platformer, etc.)), but it's a setting that can accentuate and develop mechanics just as successfully as rigid linear campaigns. The worth of this type of setting is in creating realism - the correlation between a game's concrete presentation and abstract mechanics. An open world sets the stage for experimentation and discovery and allows for spontaneous, emergent gameplay alongside technical progression. "Emergent gameplay" is easy to mistake as a buzzword for "randomness", but if the player feels in control of all the variables, it can offer the same learning experiences as the strictly guided, artificial feeling "tutorial-plus" style structure of linear games.
*it should be noted that many open world games do include closed missions - Arkham Asylum, for instance, almost always sets missions in restricted linear environments. However, this is a union of styles - it doesn't make sense to examine these missions under the banner of "open world", as they exit the open world paradigm.