I wear a pair of suede half chaps for everyday schooling, and I have a nicer smooth leather pair for shows. I find that when I don't wear any half chaps at all, the stirrup leathers pinch my legs, so I've always assumed that was their purpose. That, and to make it look like you have tall boots when you really don't
I don't go to very large shows and I have never found a pair of tall boots that fit me, so half chaps are it for me until I reach a high enough level that tall boots are required.
At the shows I've been to (schooling & local recognized dressage shows), only very young kids show without half chaps; they wear jodhpurs with paddock boots and garter straps.