Where does the Sigmoid in Logistic Regression come from?

Note: The title of this post is circular. But I use/abuse it because of the post linked below. I noticed on the Hacker News front page (and via multiple reshares on twitter), a discussion on why logistic regression uses a sigmoid. The article linked in the story talks about the log-odds ratio, and how it leads to the sigmoid (and gives a good intuitive plug on it). However, I think that the more important question is – Why do you care about log-odds?