Explanation based on classical physics, and Why is it wrong?

Basically, photoelectric effect has to be attributed to the transfer of energy from the incident light to electrons in the material. From this perspective, classical electromagnetic theory, which treats light as continuous wave, would predict that:

1,  As long as the intensity of light is high enough, electrons will be dislodged from the material, regardless of the light frequency.

2,  Changing the intensity of light would induce changes in the kinetic energy of the electrons emitted from the metal.

3, Higher intensity of incident light increases the rate of electrons being emitted.

4, A sufficiently dim light would lead to a time lag between the initial shining of light and the first emission of an electron.

Comparing these predictions with the experimental observations, one immediately notices that only (3) match while the others contradict each other. To make sense of the fact that light can only eject electrons from a material when its frequency is above a threshold, and the intensity simply affects the rate of electrons being ejected, Albert Einstein suggested that a beam of light is not a continuous wave propagating through space, but rather a collection of discrete wave packets (photons), each with energy hf. (h is Plank constant and f is the frequency). This idea of quantisation of energy was first proposed by Max Planck to explain black-body radiation. Using the idea of light being composed of discrete photons, one can explain the photoelectric effect as following.

Electrons in a material can absorb photons from the incident light, but they follow an “all or nothing” principle. If the photon energy is enough to free the electron from atomic binding, it is absorbed and electron emitted. In this case, some of the photon energy is used to liberate the electron from the material, the rest becomes the kinetic energy of the emitted electron. Otherwise, the photon won’t be absorbed (transmitted or reflected) and the electron remain in the material. Mathematically, the maximum kinetic energy E_kmax of an ejected electron is given by

E_kmax=hf-Φ

The term Φ is the work function (sometimes denoted W), which is the minimum energy required to remove an electron from the surface of a material. If we describe the work function in the form of photon energy as

Φ=hf₀

where f₀ is the threshold frequency of the material. The maximum kinetic energy of an emitted electron is then

E_kmax=h(f−f₀).

Kinetic energy is positive, so we mush have f>f₀ for the photoelectric effect to occur. This is exactly what we observe experimentally. Since an increase in the intensity of low-frequency light will only increases the number of low-energy photons sent over a period of time, it will not create any single photon with enough energy to dislodge an electron. Thus, whether electrons are emitted or not, and the energy of emitted electrons do not depend on the intensity of the incoming light, but only on the frequency (energy) of the individual photons. This also explains the fact that, even for very dim light, there is no lag between its shining on the material and first emission of electron.

Here is a Youtube video demonstrating photoelectric effect using a simple gold leaf electroscope.