I think the question is how does badminton related to memory improvement.
Well, im not a doctor to explain from medical view or a grand teacher that explain from academic view.
But i think memory is not just about remember a number, image, or sound.
There is also a cognitive memory (not sure if its the right name for it). For example:
1. When moving around in the court, i know where i stand without even see it. I feel it as i move, how big my foot step, to what direction.
2. Simillar to the 1st, now i add my judgment wether the bird is in or out.
3. Or on any badminton action (foot work, smash, drive, lift, clear), you dont even need to focus your mind on your body part & order it to do what you want. You only think what you want to do & your body will move as in your mind.
In simple its kinda habbit that we build for badminton activity.