Mishnah 2 of chapter 6 of Maseches Shevuos moves on from the law of one who admits to part of a claim, and deals with cases in which the defendant denies the claim in various ways.
The first case - an admission before witnesses:
The Mishnah opens: "Maneh li b'yadcha" - the claimant asserts that money of his is in his fellow's hands. "Amar lo bifnei eidim hen" - the defendant admits before witnesses and confirms the debt. "L'machar amar lo t'neihu li" - the next day the claimant comes back and demands his money, since the other man admitted to it before witnesses. Here the Mishnah distinguishes between two possible responses:
"Nesativ lach" - exempt: The defendant answers that he was indeed obligated, but that he paid the debt in the meantime. This claim does not contradict his earlier admission, it adds to it: what I said yesterday was true, only that since then something new has happened. He is therefore exempt.
"Ein lecha b'yadi" - liable: The defendant completely denies what he said yesterday and claims that he never owed anything at all. In this case he is liable, and even if he retracts and says that he really was obligated, and that what he said in the meantime was a lie, but that he paid the debt - his claim does not help him.
The reason for this: had he claimed from the outset that he had paid, he would have been believed, as explained above. But once he denied his own admission, he has been established as a denier - he is considered a confirmed liar, and from now on we no longer accept his claim of payment.
The condition of the admission - "Atem eidai":
The Gemara clarifies that this applies only to a case in which he admitted before the witnesses formally, by saying "you are my witnesses." If he did not designate them for testimony, the admission is not admissible and carries no weight, since the defendant can say that he did not mean it seriously but was merely joking - without designating witnesses, these are just idle words. But when he said explicitly "you are my witnesses," and afterward denied his words, he takes on the status of a confirmed denier, and even if he later claims that he repaid the loan - he is not believed.
The second case - stipulating to repay before witnesses:
The Mishnah continues: "Maneh li b'yadcha", "amar lo hen" - the defendant admits the debt. Only that here the claimant adds a stipulation: "Al titneihu li ela b'eidim" - I do not want you coming along and claiming that you paid, and therefore I stipulate that the repayment be made only in the presence of witnesses. The next day he says to him "t'neihu li", and the other replies "nesativ lach" - I paid, even though I have no witnesses to it. In this case he is liable, because he was supposed to pay in front of witnesses.
What is the source of the obligation to repay before witnesses?
At first glance this is puzzling: the stipulation was made after the loan had already taken place, and the claimant comes and demands on his own initiative that the repayment be made before witnesses. What binds the borrower to this demand?
The Ran explains that this law rests on anan sahadi - the concept that something which has become the established norm is treated as though there were witnesses to it. Since the claimant stipulated in his presence that he pay only with witnesses, and the defendant did not respond at the time and did not object, he is expected to pay in front of witnesses. This is not an obligation imposed upon him, but a determination of fact: surely this is how he acted. And for that reason, if he comes and claims that he paid without witnesses, we assume that he is lying.
In summary: In this Mishnah we learned that one who admits a debt before witnesses and the next day claims "I already paid you" is exempt, since his claim does not contradict his admission; but one who denies the debt entirely, saying "I owe you nothing," has established himself as a liar, and we no longer accept from him a claim of repayment. We also saw that an admission obligates only when the person said "you are my witnesses," and we examined the condition that a debt be repaid only in the presence of witnesses, whose force derives from the principle of anan sahadi.