Much of his letter was basically Columbus complaining about the fact that he felt he was never given the proper credit for opening up South and Central America to the Spanish, particularly because of the massive amounts of wealth they would derive from the natural resources discovered there.
He also took a great deal of time describing his actual travels in this particular voyage, again spending time focusing on the bad fortune and bad equipment that left him stranded in Jamaica for nearly a year before transportation back to Spain arrived for him.
In some ways he was continuing to try and reform his reputation with the court in Spain after having suffered so much humiliation and even imprisonment. He recounts the suffering they went through and all the hard work and heroic efforts of him and his crew as they tried to explore the eastern edges of South America.
Much of this letter, written from Jamaica in 1503, is devoted to Columbus outlining the many injustices and slights he had to endure after his discovery of the New World. He is bitter about how he was considered delusional at court for proposing his journey; how he had been promised that he would govern the New World on behalf of the court, yet now the same people who opposed him are awarded governorships; how he was taken aboard ship in chains against his will, “without being tried or condemned.” In short, Columbus is trying to show how he has been a faithful servant to Ferdinand and Isabella, and now, after his great discoveries, has been displaced in favor of those who seek only plunder. Columbus’ stated purpose in writing is to seek “the restitution of [his] honor, the reparation of [his] losses, and the punishment” of his enemies, or, short of that, permission to travel to Rome (or some other place of “pilgrimage”). But his real aim, I have to think, is to get some sort of financial restitution; after opening up for the crown the riches of the New World, Columbus himself is destitute, alone, surrounded by savages, alienated from the church, afraid, he says, “that my soul will be forgotten if it here leaves my body.”
Columbus begins his letter by mentioning titles he was given- voceroy, admiral, and governor-general. After mentioning these titles, Columbus mentions that while he was in Court, he was shunned for his ideas, but now that he has found territories, other explorers are given no grief when looking for grants to found new lands. From the letter,
"Seven years I was at your royal court, where all to whom this undertaking was mentioned, unanimously declared it to be a delusion. Now all, down to the very tailors, seek permission to make discoveries. It can be believed that they go forth to plunder, and it is granted to them to do so, so that they greatly prejudice my honor and do very great damage to the enterprise."
For 7 years, Columbus was called delusional. Now that he has opened up the West, everyone else wants to explore. Columbus thinks that by giving them no problems and by allowing them to explore, it insults his honor and generally worsens the whole enterprise of exploration.
Columbus goes on to argue that while he was waiting for accolades, a ship came and brought him back to Spain. He argues that,
"I was made a prisoner and with my two brothers was thrown into a ship, laden with fetters, stripped to the skin, very ill-treated, and without being tried or condemned."
Columbus speaks that he's young, not a grey hair on his head, and he was taken away from his service before his time. At the end of the letter, he makes a stab at the monarchs saying that he, unlike other explorers, did not sail for glory but for the Spanish crown.
So, to summarize, he wrote this letter because he wanted to try and get the Spanish crown to allow him to leave and explore other places, namely Rome. He also wants it to be known that he was the first of many, and all explorers after him are not as accomplished as he was.