Why are saddles made of leather?

Well, it is actually made of leather on a wood frame. The wood frame provides the overall shape.
Mostly because leather is readily available wherever cattle are raised and horses are commonly use to work cattle.

However the internal frame of the saddle that provides its shape is wood and metals are used both to hold the saddle together and as decoration. In special parade saddles large amounts of silver (and sometimes some gold) almost completely cover the saddle.