The phrase is often used for illnesses that are short-lived and not too severe. Sorry, I can't join you for dinner; I've gone down with the stomach bug.
Pro Tip 2/3
Colloquial Use
It's common in casual conversation, but not in formal or medical contexts. Many people in our office are going down with the flu these days.
Pro Tip 3/3
Not for Injuries
Used for diseases or infections, not for physical injuries like a broken leg. I may not attend the party; looks like I'm going down with a fever.