What is the exact meaning of the following sentence?

I think it is trying to say that you should not use things that would make your eyes roll when you hear it. For example, if a politician said:"If violent video games are allowed to persist, then everyone who plays them will become serial killers!" Personally I would roll my eyes at this because I know that this is a logical fallacy. One cannot make such as statement without rigorous evidence. We cannot claim that one event (video games) will undeniably lead to another event (serial killers), without substantial evidence.

Essentially, don't use arguments in your paper that, if a politician said it, it would be highly scrutinized.

Hope that helps? Not always absolutely imperative that you understand every sentence in an article, so long as you get the gist of it all. If you don't know a word don't hesitate to look it up as that's the best way to build vocabulary. Well best of luck!

Maybe an example would help. Suppose in a psychology journal someone wrote:" We have proved by interviewing 1549 persons in population X that 89.4 % of them would be incapable of becoming scientists.Thus we conclude that people in population X are intellectually inferior" The author knows that he could never get away with that in a political debate, but tries to spread his prejudices under the guise of scientific research [Yes, this example is caricatural:-)]

I have put your sentence in a slightly less natural order, which should make it a bit easier to follow:

In an academic paper, we can't try to get away with something that we would abhor in a political debate (or worse yet, in a political advertisement).

There is something that we would abhor in a political debate. We should not use this thing in an academic paper either; for, if we would abhor it in a political debate, so our readers would abhor the same thing used in an academic paper.

The that clause modifies something; the phrase in my example sentence in italics becomes the object of get away with.

Note that we can't try to get away with sounds a bit awkward; an alternative would be we shouldn't try to get away with, or we won't get away with, or even we shouldn't expect to get away with.

Upon reading the statement I take it to mean:

We would abhor hearing "X" in a political debate.
It would be even worse to read "X" in a political ad.
Therefore, we shouldn't write "X" in an academic paper.

Why we should feel that way isn't yet clear, but I would assume the author is about to explain...