The idea was that you could play fully with keys (arrow keys plus space) or use the mouse (hover to select, click to play, click on rotation buttons to rotate). Maybe you didn't notice the rotation buttons on the tiles when they are selected?
We aimed to have an element of deckbuilding in the game, but we didn't manage to flesh out the levels and mechanics, so with only 3 main levels and limited mechanics the deckbuilding aspect is a bit pointless.