"Penguin man" is the official First Podcast code word used by the hosts (or, occasionally, guests) to indicate, "There is a problem and we need to strike something from the record/ensure that what was just communicated does not make it onto the final product."
Generally speaking, Lars will work with whomever has made the penguin man request if it contains something unintentionally offensive, has the potential for doxxing, or otherwise would not be a good look for any and/or all parties involved.
He encourages simple foibles and mishaps to be left in, as the show is extremely casual, and the end result of "cute" mistakes is that everyone seems more relatable and human to the audience.
Usually, segments that are penguin-manned out are simply left on the cutting room floor. Occasionally, however, references to the term are left in.