In Django Unchained, Dr. King Schultz, who is a bounty hunter, takes enormous risks and goes to a great distance in order to help Django. Although Django does help him nab the Brittle brothers, was this enough reason for Schultz to help Django? I mean this was deep south in 1858 and Schultz decides to help a black man against immense odds. There must be a strong motivation for him to do that.
In the beginning, Django seem like only bounty help to Schultz but slowly Schultz and Django learn about each other and in a scene Django tells how his wife got sold in to the slavery and Schultz reveals about his wife's death. Schultz starts teaching Django how to shoot and becomes his mentor and also there were instances where Schultz admires Django's shooting abilities and says to himself "This kid's a natural."
This post says that Schultz can't stand slavery. This might also be the reason why Schultz keep helping Django to free his wife from slavery.
Schultz despised slavery and hated the sight of slaves being treated poorly and also killed. At some point in his life, Schultz was married. His wife supposedly died of an unknown cause.
On the other hand think of Schultz, he's a bounty hunter, he has no family other than the only person (Django) who he has recently befriended with. I think the reason why King Schultz go to such distance in order to help Django is because they both are friends.
In addition to the other answers Dr. Schulz tells Django in the movie why he help him: As a german Dr. Schulz feels obliged to help him to free Brunhilde after he told Django the story about Brunhilde and Siegfried. I would guess that this is a big part of his motivation, he is far from home and as others have mentioned Django is probably his only friend in a long time. Also Brunhilde and Django can speak german so that is a big plus in sympathy points for any german.