To open a slime gate you'll need a slime key. These can be obtained by popping Gordo slimes throughout the far far range. (See my Jelly Belly Burst achievement guide)
After you've obtained the key simply walk to one of the slime doors, leading to either the Moss Blanket or the Indigo Quarry, and press X to use the key to unlock the door!
This will also unlock one of the two following Achievements, depending on which door you unlock.
Here's my video for all of the achievements if you'd prefer. Skip to 5:12 for Open Says Me.