The Exotic Plants are obtained from the Seedling Kit. This Kit is granted once you progress enough through the Conservatory. You should have it on your Board, in the Inventory, or in the Pocket below the Board.
To use the Seedling Kit, you need to sink into it a Leaf Rake Level 7.

  • Leaf Rakes can be obtained from Gardening Toolboxes.

A GIF that shows how you can sink a Leaf Rake Level 7 into the empty Seedling kit and make it function.