It seems that the boy you play as is a dead one who exists only in the girl's memories. As first, the world is void of colour in the mind of the girl, who cannot let go because of her sadness. As you give her the things that she like, you give colour to her empty world. Once you help her let go of her sadness, and let go of your death, then she can finally be happy and live her life full of colour; grey no more.
Um, whenever I play a game like this on kongregate, I know there's some sort of symbolic thing, and whoever posts it gets a ton of likes, so here goes... Uh, well, so, it's a lot like real life, because uh, when it started it was, uh grey, and, er, white, and um, something. But then some dude found flowers and brought them to some random... Woman, and colors appeared, and he died because er uh um.... Yeah, colors.