I think this is basically a “shock value” interpretation of a more subtle statement. Obviously adding strictly-positive numbers does not result in a negative number under normal arithmetic. Check the numberphile video and you may be simultaneously irritated and disappointed.
Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1].
How does that work? Who can explain this to me?