Integral field optical spectroscopy using the INTEGRAL system, complemented with HST imaging, has been used to characterize the merging process giving rise to the ultraluminous infrared galaxy IRAS 08572+3915, the star formation along the tidal tails of the galaxy, and the nature of the nuclear ionizing sources deep within the galaxy. The overall morphology with two well-identified nuclei, the widespread star formation, and the unperturbed two-dimensional gas kinematics indicate that IRAS 08572+3915 is a dynamically young system formed by two disks' galaxies which are in the process of merging. The galaxies have a mass ratio of about 5, the brighter being a ~0.5L* galaxy. The ionized gas distribution traces the presence of young (6 Myr) dust-enshrouded massive nuclear starbursts of 2×107 to 108 Msolar. Contrary to previous claims, and based on the two-dimensional extinction-corrected optical emission line ratios, there is no evidence for a LINER or Seyfert-like nucleus in either of the galaxies. This is unusual for a warm, ultraluminous, infrared galaxy like IRAS 08572+3915. Tidal-induced, star-forming knots, located at distances of about 7 kpc from the nuclei and along the tidal tails, are traced by the presence of bright [O III]-emitting regions. These knots, with ages of 5.5-6 Myr and masses of ~106 Msolar, seem to represent a common phenomenon of the merging process; they are already detected in many other ultraluminous infrared galaxies, and tidal dwarf galaxies could be just the more massive manifestation of the same phenomenon. Based on observations with the William Herschel Telescope, operated on the island of La Palma by the ING in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based also on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.