Do I need to purchase tickets for Belgian Rail (Bruxelles-Brugge) before my trip or can they be purchased at the station on the day of travel?
There's no advantage for advance purchase on domestic Belgian trains. They don't offer discounts or reservations. Just buy them at the station on your day of travel.
There may be one reason to buy them in advance. Belgian rail ticket machines don't take US credit cards (but the clerks at the ticket offices do take them). If you don't want to stand in line at the ticket office, you could buy your tickets online and print them out before you go.