Why did the thief, the wife, and the man claim to have committed the murder?

This is a story in which we are given multiple perspectives and have no definitive "truth," just the testimony of various people who offer their own self-serving perspectives on the situation. I would say that in all three instances, it was a matter of honor to have killed Takehiko.

The thief, Tajomaru, claims to have killed Takehiko because he has a reputation as a brigand to maintain. While he says he had no intent originally...

This is a story in which we are given multiple perspectives and have no definitive "truth," just the testimony of various people who offer their own self-serving perspectives on the situation. I would say that in all three instances, it was a matter of honor to have killed Takehiko.

The thief, Tajomaru, claims to have killed Takehiko because he has a reputation as a brigand to maintain. While he says he had no intent originally to kill him, when Masago tells him she will go with whoever is the victor in battle, he feels compelled to fight a "fair" fight with Takehiko, and he kills him. He makes Takehiko sound like a worthy foe, to enhance his own reputation. 

Masago claims to have killed Takehiko for her honor.  She says she cannot bear to be looked upon by him now that she has been violated by Tajomaru. She says he has seen her shame, and she says she sees a look of hatred in his eyes.  Her first plan is to kill him and then herself, but after she kills him, she is unable to kill herself. Still, her dishonor will at least not be observed by her husband.

Takehiko claims to have killed himself out of honor as well.  He testifies that Masago has freely given herself to Tajomaru, and once Tajomaru releases him, he finds the sword Masago has dropped and stabs himself. To have one's wife behave in this way was a great source of shame, and this was the only path open to him.

It is important to remember that this story takes place in what I believe is medieval Japan, where honor was everything.  In most modern cultures, these attitudes and actions would not make for a credible story. But there are still some cultures in which the violation of a woman or adultery can easily lead to an "honor killing."