Save the World is not available on Nintendo Switch and Epic has said they have no plans to bring it to Nintendo's console. You'll have to purchase the game mode in one of the several packages it comes with, starting at $39.99 on most storefronts including the Xbox One, PlayStation 4, and PC. The issue is that Save the World isn't free despite Epic Game's promises to make it free-to-play.
In it, you fight zombies for as long as you can, either alone or among friends, in survival where you build and shoot until you get overwhelmed.
Fortnite: Save the World is the original cooperative game mode that Fortnite launched as before turning into the battle royale juggernaut it is today.