Because of their refusal to adhere to the King's commands, and their neglecting to sign the oath declaring King Henry the Supreme Head, both of them were tried and convicted of treason - therefore sentenced to death.
During his life, Fisher had a great reputation. He was always a religious man, and he was a rather intelligent one, too. Born in 1469, he enrolled at Cambridge University in 1483 and was ordained in 1491. He served as chaplain for the incredibly devout Margaret Beaufort (grandmother to Henry VIII), and during that time he became Bishop of Rochester and Chancellor of Cambridge, making many modifications to the university and inviting Erasmus over for lectures. Erasmus himself had quite a good opinion of Fisher, saying, "He is the one man at this time who is incomparable for uprightness of life, for learning, and for greatness of soul."
When Henry VIII was crowned King, Fisher became his chaplain and the Queen's confessor, gaining the respect and admiration of them both. The King himself boasted that no other realm held a bishop as learned and devout, and it seems that Fisher was an important and beloved member of the King's circle.
Unfortunately, everything changed once Henry began seeking an annulment from Katherine, insisting that his marriage was a sin, and citing Leviticus 18:16 - "Thou shalt not uncover the nakedness of they brother's wife; it is they brother's nakedness."
However, Fisher brought up the point that, before Henry's marriage to Katherine, the Pope had granted a dispensation on the grounds that Katherine's first marriage to Henry's brother had not been consummated, and therefore had not been a real marriage. He also brought forth another bible verse, which was quite contrary to Leviticus - this one being Deuteronomy 25:5-6 - "If brethren dwell together, and one of them die, and have no child, the wife of the dead shall not marry without unto a stranger: her husband's brother shall go in unto her, and take her to him to wife, and perform the duty of an husband's brother unto her."
But, the King was determined to get his annulment and marry his new love, Anne Boleyn. In an effort to do this, he broke ties with the Catholic Pope and instead declared himself in charge of religious matters in his realm. Of course, Catholics were appalled by this, and while everyone else was signing (some of the begrudgingly) the oath of obedience to the King, Bishop Fisher and Sir Thomas More were sadly shaking their heads. In their minds, the King had no right to do such a thing.
"I thank God, hitherto my stomach hath served me very well thereunto, so that yet I have not feared death; wherefore I desire you all to help and assist with your prayers, that, at the very point and instant of death's stroke, I may in that very moment stand steadfast without fainting in any one point of the catholic faith, free from any fear."
He was well-liked and respected during his life, and certainly did not deserve his fate. He is remembered now as a martyr for Catholicism, as Thomas More is, and is a hero to people devout and committed to their faith.
"I reckon in this realm no one man, in wisdom, learning, and long-approved virtue together, meet to be matched and compared with him." ~Sir Thomas More.