The record is held in place by a wooden peg in the center hole.   This peg is pressed firmly into a hole in the plywood backing sheet (with plenty of glue as well).   Quite simple, see.

      The recordings are :
"Out of a Million You're the Only One" by the Abe Lyman Orchestra   and
"Golden Slippers" played by H.M. Barnes & his Blue Ridge Ramblers.