Oh, weird. Sometimes in Garfield the Joke Logic gets so thick it's hard to tell if the gag is being mangled by the rules and realities of the strip, or if the writing is playing with the conventions of the same.
Panel 1: Liz, a professional who works with animals, tries to win a cat over not with treats, petting, or attention, but diplomatic conversation as if he were a respected equal. She is not without a history of speaking to Garfield, but it is usually to threaten him about holding still for shots, or as an ironic confidant for sarcastic remarks about Jon; the excuse for most other instances has been that she's talking to Garfield "as if" he understood, knowing that he does not.
Panel 2: Whether she pulled Garfield aside when Jon stepped out to the bathroom, or she has requested a moment alone with Garfield to have this important talk, the situation is so creepy, it's no wonder Garfield is frozen in disbelief and fright before discussion starts. Liz also doesn't have Jon's ability to hold naturalistic one-sided conversations with Garfield that make sense: as far as Dr. Wilson is concerned, she and Garfield spend the remainder of the strip standing there staring at each other.
Panel 3: Here's where I'm positive the Joke Logic is the joke itself; Garfield's discomfort is crazy, out of character, and silly on many levels. Not the least of which is that everyone sees him naked all the time. Not the least of which is that friends may see each other naked under all sorts of circumstances. Also-not the least of which is that Garfield's lament is supposed to parallel an uncomfortable turn in the doctor-patient relationship, but he doesn't disrobe before or share any personal information with the vet. Aw. He thinks he's people! And so the way a simple joke has confused itself becomes the joke itself.