The screenshots didn't quite make the gameplay clear but now having played it a bit it's pretty cute and clever.
-Your goal was simply to match pieces of the appropriate color/shape into their corresponding holes. Get enough in their appropriate holes and you'll complete the level with your goal being to complete the level in the fewest number of turns possible.
-Get a piece in its corresponding hole and you'll move the progress bar over to the right a bit. Work it so two or more pieces make it into their holes at the same turn you'll make even more progress progressively (2 in a single turn is worth a lot more than getting 2 over multiple turns).
-Each turn you can either move one of the catching rows one space left, right, or just hit pass to change nothing.
-If a piece finds its place it disappears otherwise it remains on the board.
-If a piece makes it all the way to the bottom of the board without finding a home you can tap it to send it back up top again (not sure if this comes at a progress bar penalty).
-Early on things are simple, not too many pieces dropping at once and only one correct shaped hole for the pieces. Later on you'll run into situations where a piece can get stuck in a whole (right shape but wrong color).