What does the quote said by Abraham Lincoln,"What kills a skunk is the publicity it gives itself." mean?

Skunks are not popular because of the terrible odor they can produce.

This can be a metaphor for a person who is publicized for the bad things they do that "kills" their career.

So the message is if you do bad things, you'll get publicity that will kill your career?

Yes, that is one way to interpret the message of the quote. The quote suggests that the negative attention or publicity that someone brings upon themselves can ultimately harm them and negatively impact their reputation or career. It serves as a cautionary reminder that drawing attention to one's own negative actions can have detrimental consequences.