What's this brush for a reuseable stainless steel mesh coffee drip filter?

It's to clean the filter. With the structure of a coffee filter, rinsing and using a towel or kitchen sponge won't be able to clean the tiny crevasses of the filter, hence the bristles are there to penetrate the tiny holes.

You would really want to properly clean out your coffee filters, as oils released by coffee could eventually build up in the holes, making your next coffee result in a bitter taste.