Why are photo electrons emitted instantly from metal surface just nanoseconds after the light falls upon it? How does the quantum theory of radiation explain it? Why can't classical physics explain this?
The photoelectric effect, which is what you are describing, is one of the basic experimental effects that forced the invention of Quantum mechanics.( The other reasons were black body radiation and the atomic spectra.)
Classically there should not be this behavior, because classically the frequency of the light should not play a role in the ejection of electrons, only the energy of the classical light beam was expected to affect the ejection of the electrons.
Nanoseconds are not "instantly". It is within the $Δ(t)$ allowed by the quantum mechanical solutions for the specific interaction : photon hitting atoms and ejecting electrons, releasing them from the atomic/molecular binding.
According to quantum mechanics The energy is transmitted by photon to electrons by the collision between the two. Only those photons can eject electrons which have energy more than or equal to a minimum required energy (threshold energy). Since energy is transferred in a lump, the ejection is instantaneous.
According to Classical mechanics the energy of light is distributed equally on its wavefront. So when light hits the surface, each electron gets a part of the total energy. So, the electron will acquire the sufficient energy in some time. So there must be a time lag between the striking of light and ejection of electron which is not true. Hence classical mechanics fails in this aspect.