The link between this song and the story can best be seen in the following line in the song:
It's a roll of the
die It's a free lunch A free ride But Nature's got
rules and Nature's got laws And if you cross her
look out! It's the monkey's paw
In the story, the White family is trying a roll of the die. The family is trying to get a free lunch. They are trying to use the monkey's paw to get things without earning them. However, nature does not allow this to happen. Nature throws in a "twist of fate" (as the song says). The Whites try to cross her (Nature) and they are punished for it.
The major theme of the story is that you cannot get around the laws of nature -- you can't get things for free and if you try -- watch out. This idea is echoed in the words of the song.