Were scenes like this one from the movie as preposterous to the average 60s viewer as they are today? My gut tells me this was primarily marketed towards children. Am I wrong? Also, if this is the wrong subreddit to ask this question, please let me know.
I think Josh Ozersky explains it best in "Archie Bunker's America: TV in an Era of Change, 1968-1978":
Basically, the 60s Batman is supposed to be funny, but a sort of tongue-in-cheek humor that plays against the predictable superhero/crime stories that came before it. The Man from UNCLE and I Spy had a similar tone, but not quite as ridiculous and campy. The over-the-top voiceovers would have been perceived as old-fashioned and funny at the time, like out of a 40s radio serial - a distinction that isn't so clear in 2014.
It primarily marketed towards children, but like Animaniacs or Powerpuff Girls, it included pop-culture references that were obviously targeted towards an adult audience as well. Here's a good example:
In the 40s, there were a couple of Batman film serials which were played straight, but they looked comically out-of-date by the 60s but became newly popular for that very reason. This inspired the 60s series, which went deliberately over-the-top since the 40s formula seemed to work. This is particular evident in the voiceover "Same bat-time, same bat-channel", just as film serials try to encourage the viewer to come back to the cinema for the next episode.