Roger Hanover asks the riddle, "You know what the mirror said to the dresser?" He then whispers the answer into Aunt Bee's ear. The answer to the riddle is never revealed, but from Aunt Bee's blush it is clear that it was bawdy. The actual answer to the riddle is "I can see your drawers."
Roger's joke about "getting over the grip" is actually a pun on the term "grippe," an archaic French word for influenza.