What does the phrase "offensively knowing" mean?

"Knowing" (definition 2b, Merriam-Webster), as an adjective, means that the person so described has inside information - one can give a "knowing look", a "knowing smile", etc.

To be "offensively knowing" means to give the impression of knowing a person's guilty secrets, and taking advantage of that knowledge to assert a higher level of intimacy or social status than would otherwise be acceptable (or pleasant).
A man who knows that a woman is having an affair might become "offensively knowing" and take liberties with her that he otherwise wouldn't dare to; a waiter or cabdriver who knows that his customer is doing something illegal might become "offensively knowing" and demand a higher tip (or give worse service) than usual.