The letter that Darcy gives Elizabeth in chapter 35 of Pride and Prejudice exists as a result of Elizabeth's refusal to Darcy's marriage proposal. Previously in chapter 34 Darcy made an awkward and ill-timed marriage proposal to Elizabeth. It is awkward because he basically explains that a marriage to Elizabeth, though made out of love, would still be considered socially inadequate. That, in itself, is insulting enough. The proposal is also ill-timed because it occurs right after Elizabeth hears Col. Fitzwilliam saying that Darcy bragged about "saving" Mr. Bingley from a bad association that would have led to marriage.
Already angry at these latter news, Elizabeth has to also endure the shocking (and badly worded) proposal. This leads to Elizabeth's rejection of Darcy not without telling him some strong words that involved Darcy's extreme sense of pride.
The result of this clash is Darcy's need to immediately write to Elizabeth a fully-explanatory letter where he tells her the reality behind her assumptions, some being correct and some being incorrect. Hence, the letter serves a pivotal role because it not only uncovers truths unbeknownst to both Elizabeth and the reader, but also because it helps to understand the rationale behind Darcy's behaviors, the reality of his relationship with Wickham, Wickham's own past behavior, and his involvement within the relationship between Bingley and Jane.
After reading the letter, Elizabeth does something she has not done in the novel before: she finally puts an end to the prejudices that she admittedly set upon Darcy and the Bingleys. Another thing that the letter does is teach Elizabeth that not everything she is told is neccesarily true; that, perhaps, it is best to watch out for Wickham whether the elopement that Darcy talks about in his letter is true or not. Wickham's sad story of neglect, abuse, and of being disowned by this "bad, rich man Darcy" may not be completely true, after all.
Here again I shall give you pain -to what degree you only can tell. But whatever may be the sentiments which Mr. Wickham has created, a suspicion of their nature shall not prevent me from unfolding his real character.
The letter explains everything. First, that Darcy was not sure enough that Jane's behavior showed true feelings for his friend. Hence, he separated them in order to avoid that his friend ends up heartbroken. In reality, Darcy is right; even Charlotte Lucas points that out to Elizabeth previously in the novel. She also thought that Jane did not look sure enough of being in love.
Also, Darcy admits to his ill-treatment of Wickham. The reason is because, after squandering the money that Darcy's father was good enough to leave for him for being the son of one of Darcy's servants, Wickham paid back by nearly-dishonoring Darcy's sister.
Finally, the letter also explains to Elizabeth something that is quite well-known to her: that it would be indeed a social inadequacy to marry her, not necessarily because of her station in society, but because of the wild and tasteless behaviors that the Bennetts display in public! This would be reason enough for any man to be concerned about marrying a Bennett girl.
Therefore, the letter is pivotal because it changes everything, not just in the plot, but in the reader's mind. Elizabeth's prejudices are now corrected and, in time, she will realize that every one of Darcy's words will prove to be true.