Here's an illustration I put together a while back that may help. The top photos show a tubular rim and a tubular tire glued to it. Notice that the surface of the rim where the tire sits is nothing more than a concave curve. The glue, which is clearly visible in the photo, is the only thing (besides air pressure in the tire) that keeps the tire from simply rolling off the rim.

The bottom photos show a clincher rim with hooks on the both edges to grab the bead of the clincher tire.