From Wikipedia, it says “Catch-22s often result from rules, regulations, or procedures that an individual is subject to, but has no control over, because to fight the rule is to accept it.” What exactly does that last part even mean? How do you manage to fight a rule by accepting it? That seems contradictory.
Yossarian: “You mean there’s a catch?”
“Sure there’s a catch,” Doc Daneeka replied. “Catch-22. Anyone who wants to get out of combat duty isn’t really crazy.”
There was only one catch and that was Catch-22, which specified that a concern for one’s own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn’t, but if he was sane, he had to fly them. If he flew them, he was crazy and didn’t have to; but if he didn’t want to, he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.
“That’s some catch, that Catch-22,” he observed.
“It’s the best there is,” Doc Daneeka agreed.”
(emphasis is mine)
So, if we return to the question raised in the OP:
What exactly does that last part even mean? How do you manage to fight a rule by accepting it? That seems contradictory.
It is indeed contradictory, and this is precisely the point of the catch - make fighting the rule impossible.
By fighting the rule, you are accepting that it is something you have to deal with, whether obeying, evading, or fighting.
If, for instance, soldiers were allowed to leave base before 9 with permits stamped on that day -- and the office that stamped them opened at 10 -- arguing that this is insane is accepting it's the rule. The only way to avoid accepting it would be to simply ignore it, and go without a permit or with one stamped the day before. Since that would be going AWOL, most soldiers would not.
Most Catch-22s do not admit of the possibility of ignoring it. You can not get a discharge by ignoring the rule that you have to be insane AND ask to be discharged as insane (which is proof of sanity).
What exactly does that last part even mean? How do you manage to fight a rule by accepting it?
The Wikipedia article does not say that you fight such a rule by accepting it, though that idea does have a certain synergy with the Catch-22 concept. Rather, Wikipedia claims that to fight such a rule is to accept it.
I don't think this is a very clear characterization, and evidently you don't either. But it nevertheless does line up with the Catch-22 concept, which is basically that there is no way out. You don't fight such a rule by accepting it. It's the other way around: you accept it by fighting it. Of course, you also accept it by not fighting it.
In the primary example from the novel, the rule would be taken to be "All pilots will fly more missions". A pilot could try to fight that rule by claiming to be insane, and they might even support that by arguing that being willing to fly more missions would be proof of insanity. Yet requesting relief from duty on the basis of insanity is taken as proof of sanity, so pilots fighting the rule (in that way) are forced to accept it.
The term "catch-22" is just a fancy name for a series of logical conclusions ending in a contradiction.
In the case of the book which coined it, the soldier in question is both sane and not sane, and this state flips around endlessly as he follows the rules laid out by his military organization.
Aside from that, it's just a plain old paradox - the same as the Barber Paradox or any other number of constructed arguments leading to a perceived or real contradiction.
But your question is about the "accepting" part. Any paradox can be resolved by deciding to ignore parts of the logical statements involved in it. For example, in the Catch-22, if the soldier ignores or does not know the fact "if he asks to be grounded due to being insane, he will be considered sane", then his decision process is suddenly very simple. There will be no contradictions and no questions about what to do, whatsoever, there will be no more paradox from his point of view. He will simply ask to be grounded because he's insane.
Obviously, as a reader, we can still know of the mentioned fact, maybe because two other characters have talked about it before in the book. In that case, for us the paradox still holds, and we can watch the drama unfold, but for the character who does not know, there would be no paradox.
So, TLDR: to "accept" the catch-22 is simply to know all the rules involved in it. To "not accept" is to not know some of the rules (or to decide to ignore them and live with the consequences; or to assume that they're false and can thus be safely ignored).