The Medrash, cited by Rashi, explains the connection between the end of VaYeira and the start of Chayei Sarah: The Satan appeared to Sarah and told her that Avraham brought Yitzchak to be sacrificed. Before he reached the story's end, she died of shock. There is another, lesser-known version of the same Medrash: In this version Yitzchak himself comes and tells Sarah what happened, and she dies from the shock.
According to the second version, in which Yitzchak himself appears before Sarah, we must analyze what it was that caused Sarah to be fatally shocked. Aviva Zornberg suggests that according to this account of the Medrash, what affected Sarah so deeply was her sudden confrontation with a stark realization of the fragility of all of our lives. In this telling of the tale, Sarah realizes that "were it not for the angel", Yitzchak would have been killed. According to this approach, the connection between Akeidat Yitzchak and the death of Sarah is the precariousness of life. Yitzchak's almost being slaughtered overwhelmed Sarah with such an unbearable existential angst that it took her life.
Today we all know of other human beings that were here one second and gone the next. Deep down, we also all know that every second WE breathe could be our last one. A lesson of the Akeida is the message that the physical world in which we exist is temporal in nature. Yitzchak's near death experience, that so shocked Sarah, serves as a reminder to us that we lack ultimate control over our lives. May we all be blessed to utilize our awareness of the delicate nature of life as a motivation to do teshuva and to constantly grow in our closeness to G-d.