The majority of nuclear waste, and pretty much all the long-lived waste, is essentially unburned fuel, with a lot of plutonium and other transuranics.
For various reasons, a liquid thorium reactor burns almost all its fuel. It's easy to remove fission products that poison the reaction by absorbing neutrons. Some of those products are gasses, which damage solid fuel and prevent you from fully using it; with liquid fuel they simply bubble out. Also it produces less transuranics in the first place since it's fissioning U233 instead of U235.
So you're able to fission most of your fuel, and your waste is just the fission products. There's a lot less of that, and in a couple centuries they're back the radioactivity of the original ore.
Regarding proliferation, the main advantage is that you're shipping non-fissile thorium to the reactor, and non-fissile fission products away from it. The reactor itself can't do much more than replace its own fissile inventory, so if a country without uranium enrichment were to start siphoning off fissiles, their reactor would shut down and they'd need to beg for help to get it restarted.
For various reasons, a liquid thorium reactor burns almost all its fuel. It's easy to remove fission products that poison the reaction by absorbing neutrons. Some of those products are gasses, which damage solid fuel and prevent you from fully using it; with liquid fuel they simply bubble out. Also it produces less transuranics in the first place since it's fissioning U233 instead of U235.
So you're able to fission most of your fuel, and your waste is just the fission products. There's a lot less of that, and in a couple centuries they're back the radioactivity of the original ore.
Regarding proliferation, the main advantage is that you're shipping non-fissile thorium to the reactor, and non-fissile fission products away from it. The reactor itself can't do much more than replace its own fissile inventory, so if a country without uranium enrichment were to start siphoning off fissiles, their reactor would shut down and they'd need to beg for help to get it restarted.