We present a theoretical investigation of the electronic structure of rutile (metallic) and M_{1} and M_{2} monoclinic (insulating) phases of VO_{2} employing a fully self-consistent combination of density functional theory and embedded dynamical mean field theory calculations. We describe the electronic structure of the metallic and both insulating phases of VO_{2}, and propose a distinct mechanism for the gap opening. We show that Mott physics plays an essential role in all phases of VO_{2}: undimerized vanadium atoms undergo classical Mott transition through local moment formation (in the M_{2} phase), while strong superexchange within V dimers adds significant dynamic intersite correlations, which remove the singularity of self-energy for dimerized V atoms. The resulting transition from rutile to dimerized M_{1} phase is adiabatically connected to the Peierls-like transition, but is better characterized as the Mott transition in the presence of strong intersite exchange. As a consequence of Mott physics, the gap in the dimerized M_{1} phase is temperature dependent. The sole increase of electronic temperature collapses the gap, reminiscent of recent experiments.