Depending on context, "let the lights go out" can mean "let a company go out of business." So, "let the lights go out on their watch" means to be the persons responsible for letting a company go out of business (e.g., by doing a bad job running the company).

The phrase is made up of two parts: "let the lights go out" means to allow the lights to shut off, turn off, or stop shining (depends exactly on the broader context). "On their watch" means under their supervision - they were responsible for ensuring that the lights stayed on, but failed to do so.