This explanation will be going by what I'm guessing will resemble this tournament's implementation based on how
APEX runs theirs.
In a tournament, there are starter stages. Players take turns striking stages until only one remains, in this case, each player will strike one stage and the remaining stage is chosen (for instance, if I strike Yoshi and you strike Final Destination, we go to Battlefield for round 1.
After round 1, we skip the character changing step since we're (I think) character locked once the tournament starts. Let's say you won the first round. We are now able to choose from all of the counterpick stages as well as neutrals (I believe). Since you won round 1, it's my pick for the stage, but you get to ban a number of those counterpicks, which I am then not allowed to choose. In APEX rules, it's 3 bans, but I'm not sure what we'll do here. Let's say for simplicity that it's 3. The counterpick list is:
Jungle Japes*
Mute City*
Prism Tower
Rainbow Road
Paper Mario*
Tomodachi Life
Tortimer's Island
Arena Ferox
I've put asterisks by three that we will pretend you banned. I am then free to pick any of the remaining stages: Prism Tower, Rainbow Road, Tomodachi Life, Tortimer Island, Arena Ferox, Battlefield, Final Destination, and Yoshi's Island Brawl. I wouldn't want to bring you back to Battlefield where you won, so let's pretend I want us to go to Rainbow Road.
That match plays out as normal, and then the process is repeated until one of us has won the match set of either 2/3 or 3/5.
If we follow "Dave's Stupid Rule", then a player cannot counterpick to a stage they have already won on. For instance, if we're in the finals and you win on Battlefield, then I win on Rainbow Road, you cannot counterpick back to Battlefield. If you win round 3, I cannot counter pick to Rainbow Road. If I win round 4, you cannot counterpick to Battlefield OR the stage you won on in round 3. This is to prevent major stage-bias from recurring in a set. This rule can be overridden if both players agree to allow a stage to be re-selected.