In this case, I don't think story is the answer to give a player motivation to clear each level. I think investing a bit of time to give the game a little more 'stimulation' would go a long way.
When clearing a level the scene simply fades into the next scene. Think of games like Sonic and Mario - when completing a level you're met with a victorious fanfare and a big 'LEVEL CLEAR' plastered across the screen. It gives the player a moment to savour their victory.
While I enjoy parts of your game like the cool teleporting animation (as well as his death animation, which I find quite cute), there's not enough 'stimulation' to keep me interested or to persevere through the challenges. Things like music and other sound effects immerse the player a little more.