The mast partners are the ship's timbers fitted between deck beams around the opening in the deck where the mast passes through - they are under the deck planking. When the mast is stepped, it is "wedged" in place at deck level, the wedges going between the partners and the mast itself. This serves to stiffen the installation, so that the mast doesn't move around and start damaging itself and the deck.
The wedges are made with a shoulder to prevent them dropping through. Not shown in this sketch (generic small coaster) is a mast collar - a ring bolted down onto the deck to take a canvas mast-boot which serves to stop leaks.