The purpose of letters in Pride and Prejudice is mainly to advance truth. While characters such as Elizabeth and others use impressions and their perspective of events to form their opinions of people's character, the letters often tear down those misconceptions and demonstrate who someone really is.
1. The most important letter is, of course, Darcy's letter to Elizabeth after she refuses his proposal. It illuminates the true situation between Darcy and Wickham because Elizabeth's false beliefs about the two men's relationship came fully from the spoken word--Wickham's. The truth lies in the written word--Darcy's. The letter in this case also forces Elizabeth to pause and "listen" to someone else, rather than hearing someone else speak but all the while forming her opinion of that person through his/her speech or manners.
2. Because letters advance truth in the novel, they usually bring characters closer together. Bingley writes to the Bennets and invites them to his house for the first time, beginning his relationship with Jane. Jane writes to Elizabeth to let her know how she has been treated when she goes to London, and this encourages Elizabeth to "take up arms" on her sister's behalf.
Austen includes an interesting modern idea in using letters in such a fashion. Perhaps as a writer, she wished to show that the written word is more effective in tearing down prejudices and pushing aside pride. Of course, being a master of writing, she would have been biased in this regards, but even our modern readers cannot argue with the fact that the written word does still have great power today. In a society of fast-talking and spin, reading someone else's writing (whether it's a letter or not) still forces someone to read and infer on his/her own. It makes us better "listeners" because we can go back and reread what we might have misunderstood, and if we're reading a letter from someone who is not absent, we're forced to use just his/her words to form opinions.