A Green Belt is an area around a city which is still largely countryside. To stop the city spreading out and getting ever bigger (something called 'sprawl') planners in many countries have decided that cities should be surrounded by a zone where very little new housing is allowed.
It is called a belt because it holds everything in. Because the area is mostly green fields and forests, this is why it has been given the name Green Belt.
Another type of green belt is a green corridor called a 'parkway'. These were designed to provide green, forested corridors alongside the main routes into and out of big cities like New York. A ring of such land, much like the British Green Belt, surrounds Washington D.C., and is called the 'beltway'.