Although Uncle Jack, himself, is a racist, why does he stand up for Atticus when Scout recounts the tale of Francis accusing Atticus of being a...

This question is a bit hard to answer directly, as it requires making logical inferences through an understanding of the text as well as the context. The scene in question comes in chapter 9, when Uncle Jack disciplines Scout for beating up Francis after he calls Atticus a "nigger-lover" (83). When the full story comes out, Uncle Jack sides with Scout, becomes immediately angry at Francis, and believes the child must be punished for what he said.

Scout stands up for Atticus here, but Uncle Jack is merely standing up for what is right.

In order to understand this, it is important to understand the context. What we currently understand and define as "racism" is any unequal treatment of people who are a different race. This was not the same definition during the time the book takes place. There were social and legal gaps between black people and white people that were not only present, but expected, and to maintain or respect those gaps--at the time--was not considered racist. It was considered normal.

This means that tension between blacks and whites was not the only racial tension. There was also plenty of racial tension between whites and other whites, as society was constantly redefining what was "appropriate" when it came to the inequality gap between blacks and whites.

While many white people at this time would not have directly disrespected a black person to his face, they might have been disappointed that Atticus was defending one in a murder trial. 

So, with that in mind, understand that it isn't so much that Uncle Jack is a racist. He is more conservative than Atticus when it comes to the societal division between blacks and whites. He is more old-fashioned, also, when it comes to progress in that area. He is obviously not out fighting for the rights of black people, but he isn't stopping his brother from doing that either. In the discussion between Atticus and Uncle Jack that follows (at the end of chapter 9) it is clear that Uncle Jack does believe Atticus is doing the right thing, despite the fact that it is very difficult. Ultimately, this scene shows that Atticus and Uncle Jack, while very different, do seem to share a similar sense of integrity. Uncle Jack, in his way, stands up for what is right in this case.