The Flower Hand is the result of Finn's arm getting ripped off by his Grass Sword after forcing it too hard to stop the platform thingy and save his dad.
Finn first got his flower hand in the episode "Escape from the Citadel," after his arm was ripped off by the grass sword and he fell into the water where some healing potion landed on his ripped off arm, causing a flower to grow there. Finn noticed his flower hand and he decided that he'd just live with his flower hand and that he actually liked it.
In the episode "The Tower," despite him saying that he'd like his arm, he got a bunch of other princesses to make new arms for him. None of the other princesses could make any arms that he wanted and then he got an imaginary arm. He planned on building a tower into space where he would reach the citadel and punch his dad in the face and then rip off his arm and replace his flower hand with it.
In the episode "Breezy," Breezy found his flower hand and fell in love with it. She decided to become Finn's friend only to get that flower. At the end of the episode, Breezy made the flower hand turn into a tree which exploded and made honey get on Finn's arm which made his arm grow back. Finn jumped for joy and cheered about this happening despite him saying that he liked his flower hand. I guess he was just happy to get his flower hand back. Oh well. Anyway, Breezy got his flower hand and kept it for herself.
The Flower Hand is Finn's arm down to a little rounded stub with a flower growing out of it. The flower has a green stem and four white petals with a yellow center.