The hagoita is a decorative wooden bat, and it comes with two "feather" shuttlecocks. (Hago means "shuttlecock" and ita means "board," thus "shuttlecock board.")
Hagoita bats are for playing the traditional game of hanetsuki, like badminton but with no net. Or, play it as a one-person game of skill. Try to keep the shuttlecock from landing on the ground!
The hanetsuki game is traditionally played at New Year, but is fun all year round.
(Not for spanking, prohibited in many jurisdictions!)