Why do stirrups sometimes have rubber pads in them?

Stirrup pads are often used - you'll almost always see them on lesson horses. They have a simple purpose - they increase the grip between your boot and the stirrup, making it less likely for your foot to slip out of the stirrup at a key moment. Racing stirrups often have integral pads with even more grip, as jockeys rely on them so much to stay on.

English style "Fillis" stirrup with pad. Image source Horlicks via Wikimedia Commons.