Indicates a strong, often sudden emotional response that can seem overwhelming. He's calm now, but mention the unfair penalty and he might go off on one.
Pro Tip 2/3
Colloquial Term
Used in casual conversation, not suitable for formal situations. In a professional meeting, avoid saying she 'went off on one'.
Pro Tip 3/3
British Slang
Primarily used in the UK, might not be understood or used by English speakers from other regions. I was confused when my British friend said his boss 'went off on one'.