The riddle
Why Did the Turkey Join the Band?
Two objects with one name
Dictionaries give drumstick two meanings. One is the stick a drummer uses to strike a drum. The other is the lower part of a bird's leg (Merriam-Webster; Dictionary.com). The riddle sets up a bird that wants a place in the band, then explains that it already owns the equipment. Its legs carry the same name as the sticks. The question sounds as if it wants a reason about music, and the answer supplies one that only works as a play on a single word.
What a drummer holds
Britannica describes a drum as a percussion instrument, one that sounds when it is struck. A drummer can strike it with the hands or with sticks. In a drum set, one player sits in front of several drums and cymbals, typically a bass drum, a snare drum, tom-toms and a hi-hat, and reaches them all with a pair of sticks and a foot pedal.
A bird in a band
A band is a group of musicians who play together, as Merriam-Webster puts it, and every band needs someone to keep time. The turkey in the joke gets the job by a technicality. Nothing in the riddle says whether it can keep a beat, and that is half the fun, since the turkey qualifies on the strength of its legs alone.
Tapping it out
This riddle gets a better reaction with a rhythm. Tap a table with two pencils while asking the question, hold the beat for a moment after "band", and give the answer on the next tap. A group can then take turns inventing an instrument for another animal and a reason it was hired.