When you open TaleSpire, nearly everything will be the same as before. The exceptions are the little seat icon on the campaign view and the Seats tab in the Settings.
For your friend to join as a guest, they will need to go to the TaleSpire Steam page and download the "Guest Edition" of the game.
Then, you'll give them an invite code, and they will use it to join your campaign.

When they do, they will get straight in if you already have seats available on that campaign. Otherwise, you will see an icon like this.

When you click it, you will either be guided through purchasing a seat or, if you already own seats, it will offer to move them to this campaign.
This also is the case if you are in the campaign already
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.