I've wondered this on and off in the last couple of years while reading or watching histories of the time. As far as I understand it, Henry and his lawyers tried to argue that Henry's marriage to Catherine was illegitimate, based on Leviticus 20:21, which in the KJV is, "And if a man shall take his brother's wife, it is an unclean thing: he hath uncovered his brother's nakedness; they shall be childless."
Naturally I can see how they might have been able to build an argument on "they shall be childless", but surely they were also far more aware than I am of Deuteronomy 25:5, which in the KJV is, "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."
Given that they must have been aware of this rather clear commandment, how did they try and pose their argument, and how did they expect it to be accepted?
(Of course, the KJV wasn't in existence at this point but I believe much of the Pentateuch in the KJV was closely based on previous translations, and I believe Tyndale's in particular so I feel safe enough with these translations - and I also rather suspect they'd have based their arguments on the Vulgate, anyway, though I'd be very interested to know if I'm wrong on that one, too.)
[Edits: minor corrections]
It was illegal under canon law for a man to marry his brother's widow (so long as the original marriage had been consummated). Marriage (and sometimes just sexual relations) gave rise to "affinity" which meant that the relatives of the spouse became your relatives in the same degree of consanguinity. i.e. Henry was claiming that as the brother of Catherine's dead husband he was considered to be too close a relative to be marriageable.
For a discussion of affinity in canon law see The Catholic Encyclopedia at http://www.newadvent.org/cathen/01178a.htm
Henry's problem in making this claim, and the official reason it was not accepted is that the issue had already been addressed prior to his marriage to Catherine. In order to get the marriage allowed he and Catherine had been required to make an appeal claiming that Catherine's prior marriage had not been consummated and so the "affinity" did not exist
However, Catherine argued Deuteronomy 25:5, what contradicted Leviticus, and with the hep of Charles V of the Holy Roman Empire, sacking Rome in 1527, and placing Pope Clement VII under house arrest, no decision could be made.' However, even though Historian John Guy argues that Pope Clement VII would have allowed Henry VIII the annulment, many people, like myself, think no because the special dispensation that the Pope made for Catherine and Henry would be juxtaposed, making him look weak, but also it is shown that Pope Clement VII used Cardinal Campeggio to procrastinate and waste time, and a decision was never made. So, arguing that Henry was not technically, married to Catherine was not real, but rather he showed his paranoia, since he thought that he was haunted by God, and it was the upmost shock for him, since throughout the whole reign he had utilised the idea of the divine right of kings (that fact that Henry was the chosen one, chosen by God)