Caution against jumping to conclusions is advised. The entire story has not yet been written or divulged. This is still a time to retain an informed skepticism.
There are rumblings - we will have to wait to see if they are 'confirmed' - that the reports from the FBI agents who conducted the Flynn interview concluded that he was not lying - rather, there is speculation (well-founded? unfounded?) that a highly-placed FBI manager improperly or illegally changed those reports to show that Flynn did lie to the FBI, in contradiction of what those agents first reported.
Why would Flynn admit to something that he did not do?
Here's one possible reason, but I can think of others:
With the threatened full coercive force of "the law" in the hands of a possibly corrupt and immoral prosecutor (who evidently is not constrained by the actual law or by normal checks and balances that are supposed to protect American citizens from abuse by the authorities), it may have been that admitting to making false statements was the only palatable choice for Flynn, even if that admission was itself a lie.
With an agreement apparently currently in place to postpone sentencing for that "admission", and other events swirling around Flynn (at least), including revelation of improprieties and likely even criminal acts having been perpetrated by people at the highest levels of the Department of Justice, the FBI, perhaps the CIA, and the Obama White House itself, isn't it a bit premature to take Flynn's "admission" at face value?
If you believe you are an innocent man being railroaded you plead not guilty and fight it out in court. Martyrdom is not supposed to be pleasant, but it is what is necessary to retain righteousness. Flynn is a twice admitted liar. He resigned because he lied to the Vice President. He entered a guilty plea because he lied to Federal investigators. If you examine the history of the sentencing Judge you will become aware that it is his custom to issue an order to disclose favorable evidence to the defense in all cases that come before him. This is a result of his experience in the Senator Stevens corruption case which saw prosecutorial misconduct go unpunished on the technicality that their conduct was contrary to case law but not contrary to any order by the court of jurisdiction. By entering a guilty plea, Flynn has confessed to the information alleged in the indictment. Having acknowledged his criminal conduct Flynn has effectively made a declaration as to the absence of any potential exculpatory evidence. As a practical matter, the opinions of the FBI agents immediately following the interview of Flynn are irrelevant. Flynn is the only person who can know definitely if he lied, and he has admitted to being a liar. Any FBI opinions to the contrary speak only to Flynn’s skill at deception which is hardly a character trait one wants examined ahead of sentencing. Should Flynn successfully withdraw his plea then his confession and all subsequent cooperation with Mueller will be admissible and will be used to convict him and the government will seek the maximum sentencing. Absent a pardon he is done. A guilty plea means a guilty man. If he feels like he copped too early that is something he just has to live with. Just like all those Bundy Ranch defendants that took pleas before the government case got shredded, they remain felons serving time. If you believe in your innocence, fight for it. If you don’t want the risk of a trial don’t complain about the consequences. Flynn is a criminal. He will likely serve time. He has admitted to willfully deceiving the country he had sworn to defend. And his actions have harmed the Trump administration. He is unworthy of our pity or mercy.