Photons carry no charge, so normally they don't interact with each other. However, at high enough energies they do. The explanation given by the best available theory is that there is enough energy that a virtual electron and virtual anti-electron form, and since those have charges, they interact, and from the outside it appears that the photons interacted.

This is totally impossible in the classical view, but is fine in the quantum electrodynamic (QED) view.