You can think of the PTBT as a big no-no finger wagging back and forth in protest.
The basic gist is pretty cut and dry: you have a nuclear bomb you want to test? Too bad, you can't do it. At least, not in the air, in outer space, or underwater because nuclear energy is wily and, like, super deadly.
With a force so unpredictable, the risks are great. As they say, if you play with fire, you're gonna get burned. Or maybe 21 miles of urban civilization will get burned. Regardless, it's a bad scene. The health and safety of people, animals, and the planet are at grave risk, so everyone needs to agree to behave and put the atomic measuring sticks away.
The PTBT is the result of careful and deliberate diplomacy carried out by the U.S. and the USSR, which paved the way for the beginning of the end of the Cold War standoff.
The PTBT was simply another insurance policy to maintain the diplomatic tensions between the U.S. and the USSR. It is a perfect example of how MAD infiltrated every level of Western society, both domestically and internationally.
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 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.
Nuclear testing is bad because it leads to nuclear weapons, which lead to nuclear war, which leads to the end of basically everything.