How do I actually play my match?

In Fortress W. When a round starts, the match is set up on the game server for the players in it — you play it in the game, not in the browser. The website is where you find out when, against whom, and on which map.

So the routine is: check your round's start time on the tournament page, be in the game before it, and play. Results come back from the game automatically; there is nothing to submit on the site.

Be ready ahead of the scheduled time rather than at it. Rounds run to a fixed schedule and will not wait.