Why does the Photoelectric Effect lead to the conclusion that classical physics cannot be all of physics?

The photoelectric effect is the emission of electrons on a charged zinc plate, when UV light is shone on it, but not low energy visible light.

This phenomena is caused by the quantisation of light. Classical physics states that waves are continuous in energy. This would mean that the electrons could be emitted if the intensity of the low energy light was high enough. However, there is no intensity of low energy light that could emit the electrons. This proves that something else is up.

This was the first piece of evidence that led to physicists thinking of wave particle duality. In this case, the waves of light act more like photons - quantised pieces of light. This experiment can be explained by particles. The electrons are bound by a certain energy, so this means that only photons of a certain energy (wavelength) can free them - no matter how many photons come into contact with it. 

This discovery was made by Einstein and along with several other discoveries in the early 20th Century led to Quantum Physics being born as a field.