They themselves live in Eternity, a place that exists outside of time. Here they have their libraries and their computers. Here they plot and analyze how to make changes with the minimum amount of disruption to the civilization they are changing. All this sounds ideal, a dream come true. Mankind is safe at last, protected from his collective stupidity and cruelty.
Mankind has a guardian angel in the Eternals.
Asimov thoughtfully examines the details of how the Eternals would work. This is, in itself, fascinating. The problem with stories about time travel is the paradox problem. If you do, then you never went back in time, never killed your grandfather, therefore he exists, and you exist so you can go back in time to kill him and cease to exist, etc. Asimov has thought of all possible paradoxes and how they could be avoided.