The English verb to have is, along with be and do, the most multifarious member of its class, with a great variation of senses and uses, including that of an auxiliary. An extended case is the use of have with postpositions. For example, to have on means ‘to puzzle or deceive intentionally; to chaff, tease; to hoax’.
The interference from this last construction’s meaning may be partly responsible for the all-but-ubiquitous contemporary use of have without the postposition on––i. e., “Thanks for having me”––in the response of interviewees to a radio or television interviewer’s expression of thanks as an opening gambit in such media conversations. But this form of the response is actually the result of a further grammatical contamination from the meaning of have without a postposition in the usage that is shorthand for invite or entertain. From the beginning of radio as a medium of communication, the postposition has been de rigueur in denoting appearance on a program. One can still hear careful speakers saying “Thanks for having me on,” although this correct usage is lamentably receding into oblivion under the onslaught of the erroneously conflated one.