Molecular bonds are made when negatively charged electrons from one nucleus are attracted to the positively charged nuclei of another atom, and the electrons are then shared between nuclei. In the case of molecular oxygen, four electrons are shared between the two nuclei, leaving eight unpaired electrons distributed equally between the two nuclei. Electrons are repulsed by each other, and the unpaired electrons distribute themselves as far away from the shared electrons and each other as possible, minimizing the energy of the molecule and resulting in a bond order of two.

In ozone, the four "shared" electrons aren't shared across two nuclei, but across three. Formally, it leads to a bond order of 1.5 and a resonant structure. Physically this means that six electrons are vying for space in between three nuclei that are packed in a bent shape. The electrons are very unhappy (high energy) to be in close proximity, but this arrangement is its lowest possible energy arrangement, filling the lowest energy molecular orbitals and allowing non-bonded electrons to spread out as much as possible. 

The high potential energy resulting from the proximity of electrons makes the bond unstable.So when ozone encounters any molecule that looks like it might vaguely like to take an oxygen atom (and its electrons with it), the molecule gladly busts apart.