To understand the phrase "See you in the funny papers" one must know a few things about American culture and history:
Newspapers used to be THE main source of all information. Everyone could afford a newspaper, or access one cheaply or free. So "the paper" (i.e. the newspaper) was the main way most people knew any fact, any trend, anything that was the zeitgeist of the time. "The papers" was equivalent to today's "trending" (i.e. "Hey, have you seen what's in the paper today? Hitler bombed The Vatican!").
In the period where this phrase came to common usage (the decades surrounding WWII) the Sunday newspaper comic strips were BIG media (and the daily strips, and the editorial cartoons, to a lesser extent). Nearly everyone read the same comics and cartoonists and kept up with what Flash Gordon or Little Orphan Annie was doing, same as we might keep up with the Kardashians. The comics pages were seen as a kind of separate part of "the paper" (often being printed in their own separate section, even) and since many strips were humorous in nature, like Popeye or Krazy Kat, this section was referred to as "the funnies" or "the funny papers."
Even though newspapers were widely available, that doesn't mean everyone had read the day's news articles or desired to. (Which is why editorial cartooning exists - if you don't know all the details of every new story relevant to today, you at least get the gist via editorial cartoons.) But almost EVERYONE read "the funnies" at a minimum. Editorial cartooning bridges the gap between hard news, in-depth articles in "the paper" for someone who, out of the whole newspaper, ONLY reads "the funny pages" or "the funny papers" and may therefore be under-informed about more important matters of the day, but who is up on what's happening "in the funnies."
And "the funnies" are usually about broad strokes matters that nearly every human being shares in common - foibles we have, humorous situations, inconveniences we all must deal with, etc. So while you may disagree with someone's take on "the news" in "the paper" chances are you both have about the same reaction to "the funnies" or whatever is in "the funny papers." So, like sports, it's an easily congenial topic to discuss amongst disparate individuals.
SO... you'll notice all of those things have to do with possible shared conversational topics one gleaned info about from the newspaper, be it "hard news" or something from the softer side, like the funnies.
If someone was having a serious conversation about weighty matters of the day you wouldn't leave the conversation with "See you in the funny papers" (unless doing so ironically). Though not a colloquial phrase, the proper sign off to such a conversation about the news of the day would have been "See you in the papers"! But if you were just engaged in light-hearted small talk it WOULD be appropriate to leave the conversation by saying "See you in the funny paper(s)."
The phrase is meant to convey a shared conviviality about current events/topics that are not mightily consequential. And to say "Hey, I've enjoyed chewing the fat with you, but I don't want to have deep, serious, personal conversation with you. I'm hitting the road now, but it has genuinely been nice talking with ya." It is equivalent to "Nice chatting with you," but with a more specific summation acknowledging the tenor of the conversation you just had.
If used ironically, of course, it is meant to convey the opposite (i.e. we ARE talking about important matters, I DO want to discuss such things with you, but I have to run now so we'll have to keep it light for now - keep it at the level of "the funny papers" - until we have a chance to have a real conversation together). Hence, you'll "see them in the funny papers" because that's the level of convo you have time for until you converse with them next time.
It is NOT meant as an insult in any way. Quite the opposite. It is meant to acknowledge a "keep it light and safe" conversation (or acknowledge the opposite among closer compatriots) designed to avoid offense. It is a way of expressing that you enjoy conversing with someone and will do so again next time you see them. Whether the phrase is meant to be understood as "Though I enjoyed it, this convo we just had was of little consequence" or "We have convos of deep import, you and I, but not right now" is, of course, dependent on how it is being used.