May Day is an in-game event, lasting from May 1 to May 7.
Speak to Tom Nook to start the event, unlocking a flight ticket at your airport.
When you talk to Orville at your airport, you'll have the option to fly to the May Day Island. You'll need 10-20 minutes to complete it. Don't worry about what to take with you to the island - your inventory will be emptied when you leave (and restored when you return).
The island contains a maze to solve. It involves moving trees, smashing rocks, crafting, chopping down trees and digging up shrubs. The maze features a special tool, Worn axe , which only has a durability of 3.
At the end of the maze, you'll find Rover, who will gift you Rover's briefcase . You can also collect 4x Bell vouchers as a reward. You can keeping talking to him, but you wont get the option to invite him to your island.
If you solve the island correctly, you should be able to collect an extra 5x Bell vouchers . To do this, you'll need to avoid moving any trees after its possible to craft the Axe , saving 3 fruit for after the maze has been solved. After reaching Rover, remove shrubs to walk west along the top of the island and around to the hidden cache, and break the rocks there. If you mess this up (and use the fruit too early), use the rescue service on your phone to reset the island.
When you return to your own island, your old inventory will be back - Rover's briefcase and the Bell vouchers will be waiting in your mailbox. Sell the vouchers at Nook's Cranny for 3,000 bells each (for a total of 27,000 bells).