Once you own seats, some extra UI will be visible on the main menu.
The boxes next to the campaign name show where the seats are, and you can click on the box to move your seats to that campaign.
When you enter a campaign, your seats are automatically moved to that campaign, so you don't have to do that yourself.
When you close TaleSpire (or go back to the main menu), the seats stay on that campaign, so your guests can still use the seat even if you are away.