Tomorrow will be Muggy, followed by Tuggy, Wuggy and Thuggy

"Tomorrow will be muggy" is a typical weather forecast.

muggy adjective (muggier, muggiest)
(of the weather) unpleasantly warm and humid:
it was a hot, very muggy evening

[ODO]

Muggy also sounds a little like Monday, which is followed by Tuesday, Wednesday etc. It's possible that someone with a speech impediment might even pronounce Monday as Muggy. The cartoon plays on that relationship and creates days in the same way: Tuggy, Wuggy, Thuggy...

It would appear that featuring disabilities is a common theme in the cartoons.

Generally muggy means being warm, damp, and close

For ex.

a muggy day in August

Reference for Muggy

But here author/writer suggest that in comical way like

muggy for Monday

Tuggy for Tuesday

Wuggy for Wednesday

and

Thuggy for Thursday.

Hot/warm days in August.