I am not sure where you got that definition from.
But if you check on the rae: desdecir
You can see the definitions in spanish.
You can see the first definition is "desmentir", which it is similar to deny. Basically someone denies something that was said or done, proving that it is not true.
The most common meaning of "desdecir" is to retract. That is how most people use it. It has a negative connotation though. It means someone said something and afterwards backed on his words, but basically because they knew they were not telling the truth. Maybe you made a promise, or said something to get some benefit. But then when someone puts you on the spot, for example they want you do do what you promised, you retract yourself. You say "I never said that" or something like that. Or "You misunderstood me".
I checked on internet and it seems that you got your sentence from this article.
Basically the guy is saying things are developing into a critical point that will force both parties to "desdecir" themselves, basically tell the truth, to deny what they have said, to their people in this case. To retract themselves.
As you can seen in his previous words
podamos retomar esa sana forma de no entendernos y de ir cada uno por su lado que tan ensayada tenemos y que consiste en decirles tú a los tuyos lo que quieren escuchar, yo a los míos otro tanto,
He is saying that each of them tells to their people what their people want to hear. Which it is not necessarily the truth. He is asking the other guy to give him something, or the situation will compromise them both and force them to deny, go back on their words: tell the truth about when they lied, tell what promises they made in vane and such.
I hope this helps, tell me if you need any more details.