Brief Summary
The Setup
The U.S. and the USSR become overly enthusiastic about detonating atomic bombs in competitive pursuit of increasingly powerful weapons. Along with poisoning the ground and the atmosphere, they scare the living daylights out of everyone. The rest of the world tells them to get a grip and put the brakes on nuclear testing.
The Text
The treaty acknowledges that the U.S., the U.K., and the USSR all have a vested interest in the further development of atomic weapons, but they agree to halt nuclear testing of certain types with the intention of eventually halting nuclear weapons development entirely.
It also invites any other nation that might be interested in joining the treaty agreement to do so (because, let's face it, you can't just move to a different town if nuclear war breaks out). However, no single nation is obligated to remain in the treaty agreement, especially if another signatory breaks its promise, i.e., runs a restricted nuclear test after signing.
TL;DR
Nuclear testing is bad because it leads to nuclear weapons, which lead to nuclear war, which leads to the end of basically everything.