There are two major factors that brought this system to and end -- both are economic and they are related to one another.
First, the open field system was not efficient. This system meant, for example, that some signficant amounts of land were wasted because there had to be spaces between the different strips so people would know what belonged to who.
Second, the system did not allow the elites of the society to make as much money off their land as they might have liked. The lord of a manor, for example, would have to allow the land to be farmed relatively inefficiently rather than making it in to one big parcel and farming it as one piece (or using it as pasture for sheep). This desire to enclose the land was another of the major causes that led to the end of the open field system.