It feels great being a Chosen One, especially after you did nothing to deserve such a status. When a prophecy proclaims you to be the One to Save the World, you feel grand, important like never before - moreso if you never tried your hand at making yourself however much important otherwise. In a story, had the prophecy never come out, the Chosen One would still be a boring, unambitious person who can only dream of being selected for such a high title.
Claiming for prophecy to "say so" and go along with it is the easiest way to show how important the main character is, even if they weren't just before - despite the unreasonability of a prophecy (which usually comes from a source that no one dares to question, a source old enough - thousands of years, often - to be completely and utterly wrong about the world) and, sometimes, the accessibility of a more prominent candidate for the role (for example, the mentor that trains the Chosen One).
Like with any trope, there are plenty of ways to work with the whole Chosenness ordeal, many of which are far more interesting. A person could simply find themselves in a bad situation and decide to do something about it, becoming a stuff of awesome whispers or even legendary in process by sheer power the radiate. Another might be trained to do a great thing, with the training reflecting just how skilled or powerful the person has to be. Yet another one might just try to survive, or protect or help someone, and the sum of the skills and abilities that they have allows them persevere great threats, often at cost of quite a few scars.
In its simplest implementation, the trope of the Chosen One is a fantasy of importance to the world, of power and fame. Immature writers - both in sense of skill and age - use this trope to project their fantasy onto a character they often associate themselves with in one way or another (age, gender, hair color, interests and tastes etc.). Some notice the projection and thrive off it (and become greatly offended if someone doesn't like their character - their interpretation of the ideal self - as a result), some write what comes to mind without thinking about it. Most often, any sort of a Chosen One is a fulfillment of dreams, ideals or wishes of some sort: Jake Sully of Avatar was explicitely chosen to be allowed into the tribe and goes on against his evil corporate-military superiors to defend the natives of the world from oppression and bloody violence. Who wouldn't want to come to another tribe, another house, another country and fix their problems and be revered as a result?
There are plenty of examples where a powerful character does great good (or evil, though those are rarer) without being once pronounced a child of a prophecy. Most recent include BioWare's Commander Shepard and Hawke of Mass Effect and Dragon Age II, respectively. Shepard was chosen into a great rank because of their immense ability (each of Shepard's chosen past tells a story of a grand achievement, so already he's not a newbie nor a weakling), but what he does, he decides to do on his own because it's the right thing to do. Hawke was an immigrant fleeing from war with his family, and protecting his family soon extrapolated upon the whole of Kirkwall.
Using the Chosenness of the One isn't a bad trope, for not only can one build a great story around, one can also twist it to get different results. Since I have nothing more to say, I take a small bow and leave the stage.