Greatly simplified, it is thus:
First, she used the "basic idea" that a "spirit" is actually an organism, composed of DNA that only "organizes" (and has the ability to do so) when it wishes. Other than that, it is as air. Considering the makeup of some fungi and seagoing colonies of life, not TOO far off. The "spirit" is basically more of a "colony" of molecules, which posseses a consciousness.
Second, (again in the novels) One such "spirit" which was attracted to human blood invaded the body of an Egyptian queen millennia ago, and fused with its DNA, causing a change in her most basic genetics. In essence, the spirit became a "symbiote" with the human, and the result was a blood-thirsting vampire. The alteration also altered the woman's (whose name was Akasha) aging process, essentially stopping it cold.
When the symbiotic lifeform (vampire) killed a victim, by sending some of her own blood back into the victim, she could also sent some of the symbiotic blood as well, thus creating a new symbiotic vampire as the "spirit's" DNA then fused with that of the new victim.
In essence, Ms. Rice attempted to explain the existance of her vampires without using anything "supernatural" or "unexplained". The result was that the reader (and the charecters) had, even in the "world" of the novels, a basic framework of rules the creatures HAD to operate under.
The "spirit's" DNA was susceptible to the Sun's rays, for example, and if it was exposed, it would die, thus killing the creature. Fire, too, was fatal. However, physical wounds were not, as the symbiotic DNA acted to heal them even as they were made. Further, the creature would die if starved of blood.
The "spirit" DNA also adapted the host in ways to make it more survivable...in addition to the longeviety, it also cured the human diseases present, and altered the human's looks to a point where they were pleasing to look at, the better to hunt victims.
Given what we now know of how parasites and symbiotic lifeforms do business (e.g., altering their victims' body chemistry), it was a good job of coming up with an explanation, which is more than some writers had done up to that point. Ms Rice, it seems, spent as much time with a science journall or two as she did creating her stories.