SKIP-BO Castaway Caper takes place on an isolated tropical island, as you've probably guessed by the name. The story gives a theme and backdrop to the card game, but doesn't deeply affect the gameplay, so don't expect too much detail. The idea is that your arrival on this island has angered the Great Volcano.
To appease it, you'll need to win 10 tribal challenges. During each challenge, you'll have to play 4 matches of Skip-Bo against the natives. You must beat every game to win.