The claim to "use nuclear waste as fuel" is often misleading. In theory, almost all reactors can reuse the waste, but it requires reprocessing.
For example, in current LWR reactors, natural uranium is enriched and put into the reactor as fuel. When the fuel is discharged, it still has some enriched U-235 in it, along with fissionable plutonium, other actinides, and fission products. If you reprocess the fuel to remove the actinides and fission products, you can reuse the fuel. The end product is called recycled "MOX" fuel and it is used by some countries. The downside to this process is cost. Reprocessing is very expensive and it is much cheaper to to just load fresh uranium into the core. One key point is that you have to remove the non-fissile actinides from the fuel because these will accumulate faster than they can be removed by nuclear reactions.
In fast reactors, you can do the same thing, except that you can load the actinides back into the fuel, so the only thing that has to be removed from the spent fuel is the fission products. This makes the final waste stream much more attractive because the actinides make up the long-lived waste, while the fission products are short-lived because they decay fairly rapidly. You can even take the spent fuel from LWRs and feed it into a fast reactor, hence the claim that fast reactors "burn waste as fuel". Note that the fast reactors can burn the actinides because the "fission to capture" ratio of the actinides is higher in fast neutron spectrums than thermal neutron spectrums.
The fast reactors also need to reprocess the fuel to remove the fission products. There are different ways to do this, but if they use metallic fuel, they can separate the fission products using a really nice electrochemical system that is much simpler than the PUREX process used to reprocess LWR fuel. Metallic fuel is most often associated with in Sodium Cooled Fast Reactors (SFRs), but it can also be used in Lead Fast Reactors (LFRs).
In fast molten salt reactors (Molten Chloride Fast Reactors), the fission products can (supposedly) be removed directly from the chloride salt, so this makes reprocessing even easier. I suspect you heard the term of burning waste as fuel from the MCFR.
Also note that actually burning waste as fuel is still very theoretical. You can get the mass streams to balance on paper, but it has never been demonstrated in practice. In practice, I suspect that you will still have to remove some of the actinides to keep the reactors critical. This is one reason why a demonstration plant is sorely needed, to test the technology.
