Honestly the best way to wrap your head around it is to try it.

In a minute I’ll DM you a Cashu token worth a few sats. The link will have you join my mint to accept the tokens. Then you can add some other mint (say https://8333.space:3338 ), transfer the tokens to that mint, then withdraw them to your LN wallet.

In that sequence of events, my mint won’t know who you are, nor will the other mint you transfer to.

In this setup you can use these LN-integrated mints to transact with other parties with exceptional privacy.