How do you get down from the mountain on cryptids island?

You don't.

You can't get in through the locked gate. So instead, you will go kite-Surfing at Cliff park. When you are attacked by Gretchen Grimlock, a Mews helicopter will rescue you from drowning and bring you to the mansion to recover.

The process

Take the $5 bill from the tree at the right of Mansion Row, and buy a sports drink at the general store. Give it to the thirsty gardener and he will loan you his shears. These can be used to cut loose the balloonist at Cliff Park (far left) and get the rope you need for the surfing kite at Kitty's Kites. When you go back to Cliff Park and surf, Gretchen Grimlock burns your kite and leaves you to drown. Fortunately a MEWS copter lands to rescue you.

Once you meet Mews, drink the glass of water he leaves for you, and go downstairs, and right, and review all of the videos in his Cryptids Museum.