What would cause a toilet to stop filling up with water?

The other answer's posted are sound as well imo, but check to see first if water is draining into the pipe where the water line to fill the bowl goes. Its usually a small rubber hose about the size of a pencil. The water level of the tank should be below the top of that pipe by a thumbs width.

The "Float Arm" can be different and integral with the fill/supply valve assembly, which also wear out over time and need to be replaced or cleaned out if possible because debris can get into it and jam it up keeping it open and water running continuously which would then spill over into the bowl filler tube. If the water level with the tank filled does not not shut off the supply valve when the float arm rises and stops its probably a worn or stuck valve or the float arm needs to be lowered so a lower level of water applies more force to close the supply valve or to stop the level below the top of bowl filler tube.