If the letter is genuine, it was to prove that Richard had no role in the assassination of Conrad of Montferrat, the king of Jerusalem.

During the Third Crusade, the queen of Jerusalem, Sibylla, died of the wave of disease hat spread through the crusader camp at the Siege of Acre. Sibylla was succeeded by her half-sister, Isabella, but that meant Isabella’s husband, Humphrey IV of Toron, would become king alongside her. The European leaders of the crusade thought that was a terrible idea, since Humphrey was a relatively minor and powerless noble. Excuses were found to annul their marriage - specifically, that they had been too young to get married so the marriage was invalid (Isabella was probably 11 and Humphrey was about 13 when the wedding took place in 1183). Humphrey was also described as “effeminate”, which could mean a lot of things…it meant that everyone thought he was unworthy of leading a kingdom, at least.

So the marriage was annulled and Conrad of Montferrat married Isabella immediately afterwards in November 1190. Conrad was much richer and much more powerful, the son of the Marquess of Montferrat in Italy and a cousin of both the King of France and the Holy Roman Emperor. Conrad and Isabella ruled from Tyre, even after the Third Crusade successfully recovered Acre - the previous king of Jerusalem, Sibylla’s husband Guy of Lusignan, still claimed the kingdom for himself, although most people considered that he lost any claim he had when Sibylla died (and that he had been responsible for losing Jerusalem in the first place). So Conrad remained in Tyre as a compromise.

On April 28, 1192, Conrad was killed in Tyre by two Assassins, who were captured and apparently implicated Richard before they were executed.

“It was standard Assassin technique to provide the killer with a misleading ‘cover story’ in order to spread mistrust and suspicion in the opposing camp.” (Gillingham, pg. 199)

So there is no real reason to believe that Richard was behind it. What would he gain from murdering Conrad? Other suspects were Guy of Lusignan or Humphrey of Toron, but murdering Conrad wouldn’t have helped their claims either. Saladin had been negotiating a truce with Conrad and it doesn’t seem likely that he would want to destabilize the whole situation again. The only people who benefited from this were the Assassins, since they were enemies of both the crusaders and Saladin.

Nevertheless, Richard was blamed, and on top of that, he had also offended the other leaders of the crusade in various ways. Philip II of France had already returned home since he couldn’t cooperate with Richard. Richard also offended Duke Leopold V of Austria (another one of Conrad’s cousins) by refusing to allow Leopold to fly his flag alongside Richard’s. Richard eventually established a treaty with Saladin to end the crusade, and he returned home as well later in 1192, but on the way, he was taken prisoner by Leopold and the Holy Roman Emperor. Their justification for arresting him was Conrad’s murder.

“This charge at least Richard was eventually able to rebut fairly successfully. He was explicitly cleared of it in a letter allegedly sent to Leopold of Austria by the Old Man of the Mountain himself. This asserted that Sinan and Conrad had a quarrel of their own, and that it was as a consequence of this, and for no other reason, that the Assassins were sent to Tyre. Although this was presumably a forgery concocted by Richard’s men in an attempt to dispel the rumours, it does look as though the message it contained, whether truce or false, got through.” (Gillingham, pg. 200)

Richard was able to convince everyone that he wasn’t responsible for Conrad’s murder, but he was still held as a prisoner in the Empire until an enormous ransom was paid for his release. But was the letter genuine? Rashid ad-Din probably wouldn't have had any desire to protect Richard, if Richard had really asked him to send a letter.

Source:

John Gillingham, Richard I (Yale University Press, 1999)