I barely understood what "deez nutz" means
something to say to annoy or frustrate something.
It sounds like the advert is telling us more than that, link is below
It seems to be an example of anti-humor, which is often seen on things like The Late Show under Letterman. (In fact, some see Dave's whole show as being a form of anti-talk show, a part of which would be the anti-humor he engaged in to parody the humor of the established form of the talk show in America). With anti-humor, the twist is in the fact that a joke or quip isn't actually funny. If you have some time, you can read a more detailed description of anti-humor here.
One could argue that an influence on anti-humor would be the Théâtre de l'Absurde, or Theatre Of The Absurd, which from the 1940's onward produced works centered on meaningless plots, repetitive or nonsensical dialogue and dramatic non-sequiturs. As with anti-humor, the entertainment value is in the sheer absurdity of it all.
Deez Nuts jokes are a humor phenomenon, like bartender jokes and blonde jokes (though not as extensive a platform), and as others have pointed out, the phrase is an altered form of "These nuts", with nuts referring to testicles.
It's always a bit tricky to explain humor, but I'll give it a try.
A young black man puts some pistachios in his pants pockets.
He is approached by two police officers.
He asks, "Y'all are L.A.P.D?" (Los Angeles Police Department)
One of the cops nods.
He says, "L. A. P. Deez nuts.", and they start clubbing him.
So the "D" in LAPD becomes "Deez" in "Deez Nuts". The phrase can also be heard as "LAPD's nuts" (The LAPD is crazy)
The Nuts in "Deez Nuts" gets conflated with the nuts being advertised.
And as far as humor goes, it's pretty labored. It depends WAYYY to much on the laugh track.