Not to be flippant, but he wrote the letter asking for clemency because he didn't want to be executed. I won't get too into the weeds on the legal process, but by the time he wrote the letter asking for clemency, he had already exhausted all of his other options. He was sentenced to death on 15 December 1961 and he had already appealed to the Israeli Supreme Court in March 1962. He and his lawyers mostly made procedural arguments, namely that Israel didn't have jurisdiction to try him for crimes committed elsewhere and that the laws under which he was convicted constituted an ex post facto judgment. Unsurprisingly, his appeal was unanimously rejected by the Supreme Court on 29 May 1962 and the lower court's death sentence was confirmed. At that point, the only way Eichmann could avoid execution was by appealing to the President of Israel, Yitzhak Ben-Zvi, for clemency, and he sent his letter to the President on the same day the Supreme Court affirmed his sentence. Other people (including his lawyer and his wife) also wrote to Ben-Zvi on Eichmann's behalf requesting clemency.

The contents of Eichmann's letter weren't publicly revealed until 2016, but it turns out that he didn't really make any novel arguments or present a compelling reason why the Israeli government shouldn't execute him. The letter itself is basically a rehash of the arguments the defense had unsuccessfully tried to make at his original trial: that Eichmann was a low-ranking functionary who was merely following superior orders, rather than acting on his own initiative, and was therefore not responsible for the mass murder of Jews:

It is not true that I was personally of such a high rank as to be able to persecute, or that I myself was a persecutor in the pursuit of the Jews ... [I]t is clear the judges in their ruling ignored the fact that I never served in such a high position as required to be involved independently in such decisive responsibilities. Nor did I give any order in my own name, but only ever acted ‘by order of’ [others].

He also reiterated the false claims that he was opposed to the Final Solution and had attempted to resign from his job in protest of the mass killings, and tried again to argue that there should be a delineation between the "leaders" like Heydrich and Himmler (who would have deserved to be executed for their actions) and those like himself, who were just following the orders of those leaders (and therefore did not deserve the ultimate punishment. You can read a non-paywalled English translation of the letter here, but, as I noted above, there aren't any novel legal arguments or new evidence, he was just rehashing the arguments from the original trial.

Obviously, falling back on the arguments that had failed to win him an acquittal at trial was a desperate strategy that was never likely to succeed, and he was really relying on public and international pressure to convince Ben-Zvi that he shouldn't be executed. Some prominent Jewish intellectuals and public figures had written to Ben-Zvi to oppose Eichmann's execution (because they opposed the death penalty, which had never been applied in Israel, not because they believed Eichmann's claims or thought he deserved clemency).

After Ben-Zvi received Eichmann's letter, there was an emergency meeting of his cabinet, led by the Prime Minister, David Ben-Gurion, to decide on a recommendation to Ben-Zvi regarding Eichmann's request for clemency. The details of this meeting were not revealed publicly at the time, and the transcript was only declassified last year (although some information was already public because it had been inadvertently leaked by an author who was unaware that the information was still classified). Ben-Gurion presented the letters from the various petitioners. Afterward, Gideon Hausner (the prosecutor) and several members of the cabinet (including future Prime Ministers Levi Eshkol and Golda Meir) spoke. Hausner defended the death sentence, arguing (among other things) that if Eichmann had been tried at Nuremberg under the legal standards applied by the IMT, he would have been executed. Most of the cabinet members, including the Minister of Justice, Dov Yosef, and Meir, who was the Foreign Minister at the time, agreed with Hausner's arguments. Eshkol argued against the death penalty on the grounds that allowing Eichmann to serve a life sentence bearing the shame of what he had done was a harsher penalty than death. In the end, though, the rest of the cabinet agreed that Ben-Gurion should recommend to Ben-Zvi that the execution be carried out. Ben-Zvi rejected Eichmann's request for clemency on 31 May and he was executed that night.

Looking at the contents of the letter, it's clear that Eichmann didn't have any compelling factual arguments, nor did he express much in the way of sincere contrition. It's pretty obvious that he knew he was guilty of the crimes for which he was convicted and his sole motivation in writing the letter was to avoid being executed. It was a longshot both from a legal and a PR standpoint, and it's not surprising that it didn't work.

Sources:

David Cesarani, Eichmann: His Life and Crimes (Vintage, 2005)

Deborah Lippstadt, The Eichmann Trial (Random House, 2011)

Hanna Yablonka, The State of Israel vs. Adolf Eichmann (Schocken, 2004)

Relevant primary sources are hyperlinked above.