The magnetic properties and electronic structure of the ground and excited states of two recently characterized endohedral metallo-fullerenes, [Gd$_2$@C$_{78}$]$^{-}$ (1) and [Gd$_2$@C$_{80}$]$^{-}$ (2), have been studied by theoretical methods. The systems can be considered as [Gd$_2$]$^{5+}$ dimers encapsulated in a fullerene cage with the fifteen unpaired electrons ferromagnetically coupled into an $S=15/2$ high-spin configuration in the ground state. The microscopic mechanisms governing the Gd--Gd interactions leading to the ferromagnetic ground state are examined by a combination of density functional and ab initio calculations and the full energy spectrum of the ground and lowest excited states is constructed by means of ab initio model Hamiltonians. The ground state is characterized by strong electron delocalization bordering on a $��$ type one-electron covalent bond and minor zero-field splitting (ZFS) which is successfully described as a second order spin-orbit coupling effect. We have shown that the observed ferromagnetic interaction originates from Hund's rule coupling and not from the conventional double exchange mechanism. The calculated ZFS parameters of 1 and 2 in their optimized geometries are in qualitative agreement with experimental EPR results. The higher excited states display less electron delocalization but at the same time they possess unquenched first-order angular momentum. This leads to strong spin-orbit coupling and highly anisotropic energy spectrum. The analysis of the excited states presented here constitutes the first detailed study of the effects of spin-dependent delocalization in the presence of first order orbital angular momentum and the obtained results can be applied to other mixed valence lanthanide systems.