I am going to try downloading the tileset again and not open it in Photoshop. Where do you generally place this in the tileset area? I placed it in the A slot but should I place it in another area like A3 or A4? Here is what I get: 
The tiles should be placed in B, C, D, or E, not any of the A tiles (those are autotiles, a special type).
If you opened the tiles in a program that affects the alpha channel (transparency) and then save the tiles, you may have lost transparency. Just try downloading again and importing directly into your game.