I don't get it. The caption on this shirt is obviously a reference to a famous--and possibly apocryphal--quip of Winston Churchill's. It's been seen in many forms ("This is the kind of arrant pedantry up with which I will not put." "This is the sort of nonsense up with which I will not put." Etc., etc.) but no matter the variant, the purpose is to make mock of the long-held superstition about not ending a sentence with a preposition--hence the ironically convoluted, terminal-preposition-avoiding syntax. Dangling participles have nothing to do with it.
Here's another prepositional anecdote: The Guinness Book of World Records once named a winner in the category "sentence with the most prepositions at the end." The honors went to this hypothetical sentence, supposedly uttered by a boy who doesn't want to be read a book about Australia again at bedtime:
"What did you bring that book that I don't want to be read to from out of about 'Down Under' up for?"
Finally, an old joke:
"Excuse me, where is the library at?"
"Here at Harvard we don't end a sentence with a preposition."
"Sorry. Where is the library at, asshole?"
