Hair brushes are actually pretty easy to clean! Clear out the hair with a comb, a rattail, and/or scissors, and then soak any gunk off of the bristles with a mixture of warm water and dish soap or shampoo. I usually let it sit for ~15 minutes and then scrub it a little against my hand, rinse it out, then let it dry completely before using it again. There are lots of tutorials on YouTube with tips for cushion-type brushes or natural fiber, &c.

You use a comb to remove the hairs, then wash the bristles and handle with soapy water. If it's a plastic or hard rubber brush, you can soak it in warm water with some Dawn dish soap. If there's styling product buildup on the bristles, a wet cleaning toothbrush with some soapy water works well. I'd try to avoid getting rubber pads and wooden handles super wet as much as possible, and don't immerse them, but they can be wiped.