What do you do with the toilet brush after you clean the toilet?
You rinse it in the bowl once the bowl is clean. Flush once or twice as necessary to rinse all the cleaner out.
We usually put the handle flat across the rim of the bowl (with the brush over the bowl) then put the seat down to hold it there while we let it drip dry for an hour or two (or until someone needs to go). Don't put the lid down - it'll smash the top of the brush and get the lid wet.
You can get a fancy little holder for it that sits in the corner of the bathroom, or you can keep a metal or plastic bowl under the vanity that you can store the brush in. Either of those should accept the brush still lightly damp and contain the drips (of what should be just clean water at this point), preventing them from spreading and causing damage.