The radial velocity of a star is the star’s velocity seen towards or away from the observer, or in this case, towards or away from Gaia.
Gaia will determine the radial velocity of its stellar candidates by looking at the way the stars emit light through the Doppler effect. That is, the frequency of light moving away from an observer decreases (redshift), while the frequency of light moving towards an observer will increase (blueshift).
This property can be accurately determined by studying high-resolution spectra of stars, by looking at the way the spectral lines move in the spectrum as the star moves. For example, the spectral lines will appear to move towards the red end of the spectrum as the star moves away from Gaia, while a star moving towards Gaia will have its frequency of light compressed, shifting the spectral lines to the blue end of the spectrum.
Stellar radial velocities are needed to determine the velocity of stars with respect to the Sun, and are therefore essential to our understanding of the kinematics of our Galaxy.