From the wiki, the idea comes from an essay that somebody has written about a conversation they had with a friend about the struggles of chronic illness. The conversation took place at a restaurant, and she grabbed the spoons for use in a metaphor because there were spoons nearby. She gave her friend a set of spoons, and every time her friend mentioned doing a task, she took a spoon away.

It could have been anything, but spoons happened to be at hand and she wanted to make a physical representation of an abstract concept. The essay resonated with people, so spoons became entrenched. And now I hear people say that they're all out of spoons to express the idea that they've done all that they can that day.