I'll look at the issue from an urban planning perspective because this seems to be what the OP implied. There are other, equally important uses of green belts (such as ecological corridors, shelterbelts to protect against sandstorms, windbreaks to protect farmland...). Built around urban centres, green belts can absorb carbon dioxide and practise transpiration, thereby alleviating the urban heat island effect. They provide important recreational space for residents and improve the quality of the air and thus the living environment. They are visually appealing as well. The image on this page shows stretches of vegetation cover around London, and this is highly encouraging; all cities and towns should have as large an extent of greenery as they can afford to provide without encroaching on the amount of land or financial resources for other important projects.
Urban sprawl is a problem. Cities are getting too big and the amount of nature is our world is rapidly decreasing. We definitely need a way to control this, and though it isn't perfect, at the moment green belts seem to be the answer. I understand that the population is growing, and that some people have no choice but to build informal settlements, but for the time being, we have plenty of brownfield land, and houses that aren't lived in. Governments really need to try and house everyone is none greenfield areas, so that we all have homes, before we can find a replacement for greenbelts, however, for now, it's our best shot.