The Gaia mission of the European Space Agency (ESA) has been routinely observing Solar System objects (SSOs) since the beginning of its operations in August 2014. The Gaia data release three (DR3) includes, for the first time, the mean reflectance spectra of a selected sample of 60 518 SSOs, primarily asteroids, observed between August 5, 2014, and May 28, 2017. Each reflectance spectrum was derived from measurements obtained by means of the Blue and Red photometers (BP/RP), which were binned in 16 discrete wavelength bands. We describe the processing of the Gaia spectral data of SSOs, explaining both the criteria used to select the subset of asteroid spectra published in Gaia DR3, and the different steps of our internal validation procedures. In order to further assess the quality of Gaia SSO reflectance spectra, we carried out external validation against SSO reflectance spectra obtained from ground-based and space-borne telescopes and available in the literature. For each selected SSO, an epoch reflectance was computed by dividing the calibrated spectrum observed by the BP/RP at each transit on the focal plane by the mean spectrum of a solar analogue. The latter was obtained by averaging the Gaia spectral measurements of a selected sample of stars known to have very similar spectra to that of the Sun. Finally, a mean of the epoch reflectance spectra was calculated in 16 spectral bands for each SSO. The agreement between Gaia mean reflectance spectra and those available in the literature is good for bright SSOs, regardless of their taxonomic spectral class. We identify an increase in the spectral slope of S-type SSOs with increasing phase angle. Moreover, we show that the spectral slope increases and the depth of the 1 um absorption band decreases for increasing ages of S-type asteroid families.