Nuclear poisons have built up in the fuel rods a bit like rain coming down on your bonfire, leaving plenty of fuel behind but it's 'wet' and mixed in with dirty ash and bits of rock and last night's beer bottles all melted into it. But if you wash off the ash and junk and dry out the logs, you can eventually put them back in a future fire. That's what a breeder reactor does. It usually involves a radioactive chemical bath tub that melts down the fuel and siphons out the good stuff from the bad. The good potential fuel is put back into a fuel rod and then left around the edge of a reactor for a year to 'dry off'. (It needs to soak up some more neutrons to be ready to fission again.) Of course, it's a lot more technical than that, but that's about the level of story that my brain likes to process things at! Breeder reactors are special bonfires with a spot around the edge that dries off the wet firewood. Easy! ;-) Anyone interested can watch this 4 minute Argonne Labs movie.
https://www.youtube.com/watch?v=MlMDDhQ9-pE
There is an even safer method of doing this that bypasses the 'washing off' stage, which has some (in my view very overblown) proliferation concerns. That's the MCSFR, a reactor you can just dump the 'dirty wet firewood' straight into but is so hot it will burn up the wet firewood and any junk on it all in the one go.