The scene, watched in a vacuum, seems pretty straight forward. Scarlett is begging Rhett not to leave and telling him how her life will be in shambles. Rhett replies (evilly) “Frankly my dear, I don’t give a damn”
The background, of course, is much more complicated. Rhett had already given up on the relationship, thanks to Scarlett’s complete and utter apathy towards him (despite the fact that he made several sacrifices for her benefit). The final straw is when Scarlett runs into Ashley’s arms for comfort after Melanie’s death, with Rhett standing right there.
Now the book, in my opinion, portrayed the scene much better. Rhett talks to her for a minute, telling her how he used to love her but now he feels nothing. How he tried and tried to feel for her and wants to be able to care about the relationship, but (takes a breath and pauses) “Frankly my dear, I don’t give a damn”.
It’s a very sad and disheartening line from someone who was telling the truth. Not some callous taunt delivered to break a persons heart.