A star is born with a finite amount of Hydrogen. When all of the Hydrogen in the core runs out, the nuclear reactions at the core stop and then there is no longer anything to balance out the gravity and the core begins to collapse in on itself.
The heat generated from this heats up the Hydrogen just outside the core. This is enough to initiate fusion reactions outside the core in the star's shell. Also, inside the core, Helium begins to fuse to create heavier elements. The increased temperatures lead to an increased reaction rate which leads to higher amount of energy released. This leads to the star becoming bigger and increasing luminosity.
Because the surface area is higher, the temperature spreads out more, meaning that there is a lower surface temperature. This causes a shift in the visible light output towards the red end of the spectrum.
Red giants only form if the star has a solar mass from 0.5 to somewhere between 4 and 6. 1 solar mass is the mass of our Sun.