Adjectives describe something in more detail. A green car. A first job.
A compound adjective isn’t any fancier, it just takes two words to do one job. A blue-green car. Some first-hand experience. Really, we intend it to be a single descriptor—one word, really—so we use the hyphen to make it so. Without the hyphen, the reader can me misled: is a wild animal trainer someone who trains wild animals? Or an animal trainer who goes crazy at parties? Calling her a wild-animal trainer tames the sentence. It’s really a single description: what kind of trainer? The wild-animal kind.
So if you see a man eating chicken, ask for a bite. If it’s a man-eating chicken, you’re the one who’s going to get bit.