What do we learn about Scrooge's home life when Fan visits him at school?
We learn that Ebenezer isn't a part of it much, as he is away at boarding school. We also get the impression he might have been there for a while. When Fan comes to retrieve him she says that their father is "much kinder now," so much so that she thought she would try one more time to ask him if Ebenezer could come home for Christmas. This indicates that she has asked before and has been refused, which further implies there may be some strain in the relationship between at least father and son, if not the whole family, until recently.
Although the change in Ebenezer's father is not explained, it is interesting that Ebenezer himself becomes so cruel and harsh later in life only to also suddenly change right at Christmas, just as his father did.
In addition to the small amount we can glean about his father, we can also see that Ebenezer and his sister are very close. Although she is much younger than he is, they obviously adore one another, as we can tell by the way they interact:
"She clapped her hands and laughed, and tried to touch his head; but being too little, laughed again, and stood on tiptoe to embrace him. Then she began to drag him, in her childish eagerness, towards the door; and he, nothing loth to go, accompanied her" (Stave II).
He goes on to agree with the Ghost of Christmas Past when he says she had a big heart.
From this Stave we learn nothing of the rest of Ebenezer's family, including his mother.
The Ghost of Christmas Past has taken Scrooge back into his childhood so that he might see where his loathing of Christmas originated. Scrooge's father was apparently a mean man, and young Ebeneezer resented being sent away to a boarding school. When his sister, Fan, arrives to bring him home for Christmas, she tells him that their father "is much kinder now". This suggests that there might have been some abusive tendencies on the part of Scrooge's father.