very nice game. i like the concept.
for such a game you won't need any special libraries. i would do it just with a canvas and types objects. this is similar to a game i did some time ago (
Tiny Puzzle) or this one that is more complex (
Power Blocks)
the concept is the same. you puzzle parts and a puzzle board and you drag the parts to the board.
it's more simple than it looks. all your parts are basically even rectangles. small rectangles that in specific combination create a shape. when you drag the shape you need ON THE RELEASE PART event to check if the grid rectangles are empty and you can put the shape in that place then you check if a line is completed or any other function you decide.
have you made any games already? you will need some basic understanding to create such a game.