So here you go, I'll give you the bare bones of the story as it was told to me and then a bit of an alternate that I have heard since. A bit of gore at the end so keep that in mind. In this story you are taking apart the bleeding heart flower bit by bit and using each bit to tell that part in the story.
Once upon a time there was a beautiful princess and a handsome prince (were there any other kind of monarchy?). The prince loved the princess and wanted to marry her. He asked, she said no. He figured if he gave her some pretty fantastic gifts then he would win her heart.(big, big red flag, not worthy)
He gave her some bunnies (normally they would be pink but alas, no pink bleeding hearts left here- so white will have to do -no wonder she wouldn't marry him - plain old vanilla bunnies? not very original)She still said no. So he gave her some beautiful dangly earrings.
Still, no go. Gave her some lovely slippers.
Nope, she's not buying it.
So, since he could not have the love of his life, he took his sword (that's the green bit - you have to use your imagination a little) and stabbed himself in the heart.
There you have it - how the bleeding heart got his name. Of course I like to jazz it up with a bit of flourish when I tell people in person. ;-)
O.K and now the newer version I have heard. Same story 'til you get to the earrings - no earrings in this story, skip that bit and go to the slippers, save the earrings to the side. Now when you get to the stabbing part you show the part you used as earrings like thisand they make a heart shape. I quite like that bit, actually. So here is a pic of all the bits.