Highly monochromatized electrons (with energy distributions of less than 30 meV FWHM) are used in a crossed beam experiments to investigate electron attachment to oxygen clusters (O-2)(n) at electron energies from approximately zero eV up to several eV. At energies close to zero the attachment cross section for the reaction (O-2)(n) + e -->(O-2)(m)(-) (for m = 1, 2, and 3) rises strongly with decreasing electron energy compatible with s-wave electron capture to (O-2)(n). Peaks in the oxygen attachment cross sections present at higher energies (approximate to 80 meV, 193 meV, 302 meV) can be ascribed to vibrational levels of the anion populated by attachment of an electron to a single oxygen molecule within the target cluster via a direct Franck-Condon transition from the ground vibrational state v=0 to a vibrational excited state upsilon' = 7,8,9,... of the anion produced. The vibrational structures observed here for the first time can be quantitatively accounted for by model calculations using a microscopic model to examine the attachment of an electron to an oxygen molecule inside a cluster. This involves (i) molecular dynamics simulations to calculate the structure of neutral clusters prior to the attachment process and (ii) calculation of the solvation energy of an oxygen anion in the cluster from the electrostatic polarization of the molecules of the cluster. The occurrence of this polarization energy at the surface of larger clusters explains the appearance of an s-wave capturing cross section at 0 eV and the slightly smaller spacings (compared to the monomer case) between the peaks at finite energy, as observed experimentally. The relative transition probabilities from the ground state of the neutral oxygen molecule to the different vibrational levels of the anion are obtained by calculating the corresponding Franck-Condon factors thereby resulting in a reasonable theoretical fit to the observed yields of negatively charged oxygen molecules and clusters.