p1. There's a subtle violation of cartooning conventional wisdom in the first panel. Time flows in a fluid, dynamic manner through comics panels when they don't depict a moment of frozen temporality; i.e.- any time you're reading dialogue, time is passing in the panel. The period depicted in the first panel spans the time it takes for the doorbell to ring, Garfield to physically react, and Jon and Liz to have their exchange. This time-crammed panel isn't unusual, but the left-to-right reading rules of the Western world usually demand that the incidents unspool in an organized fashion that mimics our reading flow - left to right, or top to bottom - so that the causality is clear. The typical way to lay out the panel would be with the eighth note / DING DONG on the left, or hovering above the reactants. The front door is more often than not to the right of the kitchen and dining room, but the geography of Jon's house is malleable, so the action could easily be staged with the cast reacting to a doorbell off left. The solution may not be a Scott McCloud approved method of seamlessly depicting time-passage, but isn't entirely a botch-job. Because the huge sound effect takes up 1/6 of the panel with jaunty lettering, it is likely to draw the eye before the dialogue, and before we begin studying the characters who we've seen in these poses roughly 10 bazillion times.
p2. Panel 1 is going through such layout tortures to preserve a left-to-right line of action, because it has to extend the imaginary stage into off-panel space. The fast/slow area dynamic is being played with, as our eyes push through left/right, only to find static images of people and cats standing around (slow area), with their attention and eyelines focused on off-panel action to the right. The gag is that the pizza delivery person is fleeing Garfield, presumably running and screaming (fast area): the structure of the joke, of the strip, of the composition (even the house's siding is angled to slide us along the path) tug our eyes continually right, playing with the reality of how we physically read Garfield. The strip simulates the urgency of the delivery person running away, even while remaining glued to the spot to watch Garfield and Liz's reactions. The imagination fills the voided image of the delivery person and the flight from Garfield. The joke doesn't hinge on that terror, but on Garfield's nonchalance and Liz's natural surprise and shift to understanding: this happens all the time.
p3. The dust eddying back in Garfield's face is the last element of a sure hand using eloquent cartooning shorthand to build a story in which the action takes place in imaginary space, centering on a character we never see.
The punchline, however, doesn't make a terrible lot of sense, because "customer appreciation" normally means a show of appreciation for customers, not, as Garfield has it, an overzealous appreciation by the customer.
In other news, winter is apparently over in Indiana, and the shrubbery have returned in bushy green force. For the Lizes of the world, this may signal Short-Sleeved Sweater season, but our fashionable readers are advised never to wear such a garment, even in unseasonably warm weather.