Nukes pretty please: What's so good about thorium nuclear reator fuel?

Availability. Thorium is more plentiful in the earth's crust than uranium (about 3, or 4 times more so). Thorium is 400 to 500 times more plentiful than uranium-235 (which is currently the only fuel we really make use of).
It doesn't even need to be mined because it's available as waste from the tailings of many rare earth mines. The kind of rare earth mines needed to make wind turbines, and solar panels. There's so much of this waste, we make about 40 times more, per year, than we need to power the world with!
It does not need to be enriched like uranium, saving considerably on cost and energy.
The thorium fuel cycle can begin using a blended mixture of plutonium and thorium. Such plutonium can be made from nuclear bombs or reprocessed 'spent nuclear fuel'. Britain has 140 tonnes of such plutonium stored, waiting to be disposed of.
Thorium is then bred to make uranium-233
In the thermal neutron reactor range, U-233 has superior neutronics to any other fissile material:

Thermal cross-section (barn) ratio of capture+fission neutrons to capture neutrons

Scatter Capture Fission

U-233 12 45 531 12.8

U-235 10 99 583 6.9

Pu-239 8 269 748 3.8

With U-233, only one in 12.8 neutrons are wasted. In comparison, 1 in 3.8 Pu-239 neutrons are wasted.
When Pu-239 absorbs such a neutron, it makes transuranic waste which can't be fissioned, and has a half-life of hundreds to thousands of years. This is radiotoxic, and is a reason reprocessed spent nuclear fuel, called MOX, is so limited - the more it's reprocessed, the more transuranics, the more radiotoxic it becomes. Transuranic waste like this can only be disposed of with time, or a fast reactor. Thorium fuel can be reprocessed avoiding this pitfall. It will make only a small fraction of the transuranics made in Pu-239, and U-235 powered reactors.
Even when U-233 absorbs a neutron (capture) to make U-234, such U-234 can absorb a second neutron to make U-235, at which point it gets another chance to fission.
Because so little U-233 is wasted, thorium/U-233 has by far, the best waste profile of any fuel type. It makes the least possible quantity of transuranics.
A LFTR is a molten salt reactor using thorium fuel. The fuel can be denatured to protect it against the possibility of diversion for weapons making, so making the LFTR proliferation resistant.
Some of the spent fuel waste is actually valuable. For example it contains considerable amounts of rhodium, one of the rarest elements on earth, which has specialist applications in chemical catalysis.