How long does the red giant phase of a star last?
Red Giants typically have a life span of only a few million years.
A few million years in terms of the life cycle of the stars is only a short period of time.
The reason it only lasts for a few million years is due to how the Red Giant uses its fuel in the core:
But first things first a Red Giant is formed when a main sequence star such as our sun runs out of Hydrogen to fuse into Helium. The loss of Hydrogen causes the core to collapse inwards, which increases the temperature of the core due to increased pressure. This increased temperature allows the fusion of Helium into what will eventually be Carbon.
Life Cycle of a Main Sequence star showing Red Giant formation (pathway on the top of the picture):
Image from: http://www.seasky.org/celestial-objects/stars.html
The temperature becomes so hot in the core that the fusion of Hydrogen occurs in the outer layers of the sun, whereby the formed Helium falls back into the core, which is used as fuel to form Carbon.
While this process is occurring the star becomes larger, due to Carbon having a heavier mass than the components that made the Carbon. The increased mass and loss of Helium fuel then destabilizes the star after a few million years and the star sheds its outer layers in what is known as a planetary nebula, leaving a white dwarf star behind.
Hope I helped :)