Think of personal liability as following you, not your house. It covers the insured's personal, non-business activities wherever they happen, as well as conditions at the home.
Typical covered exposures:
A dinner guest trips on a loose rug in the hallway.
Sparks from the insured's barbecue burn a neighbor's shed.
The insured's dog knocks down a jogger in a public park.
On vacation in another state, the insured accidentally hits another golfer with a ball.
Coverage applies when the insured is legally liable for the injury or damage. Careless (negligent) conduct is the classic covered situation. An insured's admission of fault does not by itself create coverage; what matters is legal liability.
A common trap is thinking Section II is limited to the residence premises. It is not. Where the accident happens matters far less than what the insured was doing: a personal activity is generally covered, a business activity generally is not.