Whenever someone dies in Tanach, there is generally one line to explain how they died, at what age, and where they were buried. Sarah's death is unusual in that it is followed by a whole chapter telling every detail of the process by which her burial cave was purchased and her buried in it. Not only is this description very unusual in its length and subject matter, but most of the details in it seem rather superfluous. Why did they need to be mentioned?
Offhand, I can think of four lessons we learn from the long description.
1) This is the first time that Avraham buys land in Israel for his personal use. (He may or may not have bought the land his altars stood on, but even if he did, the land was probably considered like "Temple property" and not available for private use.) This indicates the permanence of his settlement in the land, and the fulfillment of the blessings related to the land. Burial caves were designed for use by many generations, so it indicates that his descendants would be buried (and live and die) in Israel as well.
2) The people of Hevron demonstrated their respect for Avraham: "Bury your dead in the choice of our graves". This respect is a fulfillment of the promises God made to Avraham, beginning with "I will bless you, and magnify your name" (12:2).
3) The people of Hevron called Avraham a "nasi elokim... betochenu" - a religious leader among them. Avraham's efforts in building altars everywhere, praying publicly, behaving morally towards other people, and so on were apparently not in vain. While the people did not reach the religious level of Avraham and his household, they were certainly better off than they were beforehand. They now recognized the good even if they did not live up to it.
4) The people of Hevron wanted Avraham to bury Sarah in their burial caves. But Avraham insisted on purchasing and using a separate cave. He was respectful to them, but kept himself distinct from them. He was not willing to dwell together with Canaanites in life (i.e. Yitzchak's wife) or death (in burial). At the same time, he bowed down to and showed respect for them. Perhaps this is a model for our interactions with people whose way of life we don't agree with, but which we cannot hope to quickly change.
To summarize: we learn that the promises to Avraham were fulfilled, and we gain more examples of Avraham's good behavior to learn from.