The “skinny” on something is the real story or information. “Getting the lowdown” on something means the same thing. Both are very informal expressions. 😄

As Ian said, "the skinny" is an idiom that means "a summary of the important details." The preposition "on" here means "about [a topic]." The topic could be anything. Here, it is obesity. It seems that this is probably a headline for an article, where the writer was trying to be clever by using an idiom that can also mean the opposite of "obese." It's what we call "a play on words," which means that additional meanings or humor are hidden within the words. The headline could have been "What You Need to Know about Obesity." But this would not have been clever. Often, headlines try to attract more attention with emotional words or clever word play. So this writer chose "The Skinny on Obesity" as the title/headline.

In this case, "on" means "about." This sentence is a slang expression that means "The important information about obesity."