As far as how it stays on, you'll see the strap in the back that helps this Chief Warrant Officer's Campaign Cover stay on. However as enthusiastic as Marine Corps Drill Instructors get I have seen on more than one occasion the hat falling off.
It dates back in US Service to WWI and in Modern service is worn by The Drill instructors, Marksmanship Instructors and Range Officers in the Marine Corps, Male Drill Sergeants in the Army and Air Force and all Coast Guard Recruit Company Commanders.
Interestingly enough, Female Drill Sergeants in the Army and the Air Force wear the Australian Bush Hat. It just doesn't look as intimidating if you ask me.