There is a common misconception that the offensive part is the tar baby scene in the animated segments because "tar baby" was a common racial slur for a long time. Indeed the idea of a tar baby character is still rightfully offensive is to many people (although actually deeply rooted in an ancient origin myth from Senegal).
In actually, it's the presentation of the live-action frame tale is full of outmoded,, fictional, embarrassing racial stereotypes that many people, black and white, considered outdated and offensive when the film was released in 1946. How to deal with race was hotted debated in Hollywood since before Gone with the Wind. But after WWII and the holocaust, there was a giant effort to try and balance for more realistic portrayals and squash stereotypes.
The fact that many don't recognize how offensive the film is at times is both a positive and a negative. On the positive side, many people don't recognize the racial stereotypes because many of those stereotypes are now thankfully obscure...on the negative side, many people still don't realize how inaccurate and insulting those characters are.
The actual stories of Lapin and Bouke that inspired Brer Rabbit are a huge part of oral storytelling tradition and well worth reading the stories and studying their history.