Nothing changes for most shoppers: no puzzles, no checkboxes.
Google's small reCAPTCHA badge appears in a corner of the page.
A shopper Google scores as a likely bot gets a message that verification failed and can't log in or register at that moment. Trying again a little later, or from another network, usually works.
Shoppers who block Google scripts (some privacy extensions) may be refused. Ask them to allow the store's scripts.