The purpose of handlers in non-MVC projects is to provide some type of encoded response, outside of HTML. Typically, a handler would return XML (rss, RESTful, etc), JSON for jQuery or other Javascript, or sometimes just pure data such as file binary downloads. I've used handlers to even return special javascript to be excuted on the client, as a way of lazy-loading large binary or requirements on a "demand-only" approach. More or less, a handler would be used to return "anything but HTML".
In MVC, you would move away from handlers and utilize the Controller to return whatever data you like. So, in the method like:
mywebsite.com/restapi/content/56223
You RestfulContentController would have a method for Index(), that would NOT return a View(), but instead pure XML or JSON.
public class JSONContentController : Controller
{
public JsonResult Index(int ContentID)
{
// get Content() by ContentID
//
// return a JSON version
return Content().SerializeToJSON();
}
}