This story lends itself well to simplification, and to the understanding of children. The first thing that might be helpful is to reduce the story to its most basic events, and the characters to their primary traits. The main events of the story are an exposition, or introduction of the scene and the characters. It is set in Greenwich Village, during winter, and Sue and Johnsy are artists that live together in an apartment. That is the introduction at its most bare-bones minimum. That will help to set the scene for your children in a simple way. Maybe a brief description of the neighborhood, and of Sue and Johnsy's relationship and fondness for one another. Next, the first conflict in the plot is introduced: pneumonia. To bring this into a story for children, you could change the name (pneumonia is hard to read and pronounce) and generalize it as a sickness. Or, you could change the illness itself to some sort of monster or creature that is a more concrete threat. The next conflict is that Johnsy gets ill; Sue, concerned, brings Mr. Behrman in. You could give a brief physical description of him, and describe him as a grumpy man with a soft heart.
Describe Johnsy's conclusion about the leaves on the vine, and Sue and Behrman's concern. Then, maybe, instead of describing Behrman's involvement at the very end, describe it in real time as it is happening--this is less confusing for children, who can't grasp timeline jumps quite as well. THey can understand what he did and its impact as the story goes along.
I hope that those thoughts helped; take the basic events of the story, simplify and condense them, and put them in chroniclogical order. That would be my suggestion for helping children understand this story better. I hope that helps; good luck!