Why don't people on the opposite side of earth fall off

Well, for one, gravity keeps you on the ground. The force of it will not let you just float away.

Secondly, let's say there wasn't gravity. It would still be impossible to "fall off" the Earth. The Earth is not flat/square - you cannot just walk off the edge. If there wasn't gravity, you would end up, after a really long time, float up through the atmosphere into space. However, you would lose too much oxygen and would die before you got there.

People all over the earth are attracted to earth by the force of gravity.