In A Christmas Carol, how does Scrooge find out that happiness comes from caring about others and being cared for?

Primarily, Scrooge learns this lesson from his observation of the Cratchit family. When he is observing the family with the Ghost of Christmas Present, for example, Scrooge watches the family with interest. Although they have very little in material terms, their love and compassion for each other are very clear. Instead of focusing on what they don't have, the family enjoys each other's company and blesses everyone, even Scrooge himself.
Scrooge is clearly very affected by the behavior of the family. He asks the ghost, for instance, what will happen to Tiny Tim and begs that Tim will be "spared" to spend another Christmas with his family.
The example of the Cratchit family, therefore, provides strong evidence to Scrooge that Christmas is all about family love and affection.
In addition, we see further evidence of this idea in the final stave of the story: Scrooge devotes the rest of his life to supporting the family and being a "father figure" to Tiny Tim.