A climbing brush is a brush for quickly cleaning grips. Climbing holds can give less grip due to residual climbing chalk, for example. Due to the layer of climbing chalk or other dirt, handles can feel slippery. Brushing is not only done on the handles for your hands, but also for your feet. Climbing shoes also leave rubber on climbing holds, so that extra layer can make the surface slippery.
Brushing is done both indoors and outdoors on rocks. In bouldering halls you often find brushes (elongated) to clean handles. You can also use your own climbing brush. Climbing brushes come in all shapes and sizes. Wood is often used for the handle and boar bristles for the brush. Nylon hairs can damage handles. Small and narrow brushes are used for crimp handles and large brushes for demolition because of the cleaning surface.
Climbing brushes can wear out just like shoes. Due to the sanding surface of climbing walls and grips, the brushes are getting smaller and smaller. As a beginner you often do not need a climbing brush. When you are advanced and the grips become more difficult, brushing the grips can make a lot of difference.