The Problem
In a metal there are electrons, which is very loosely tied to the metal, and when you shine light of the metal the light will knockoff electrons. This means that the light gives of energy to the electrons in the metal thereby making it possible for them to escape. And the more energy the light has, the more energy the knocked off electrons will have. This was known, but it was thought that the energy in a light wave depended on two things:
1. The intensity (amplitude) of the light wave.
2. The frequency of the light wave.
These two together decided what the energy level of the wave would be. That means that if you have a wave with certain energy you could turn down the frequency and turn up the intensity and you would have a wave with the exact same energy level as your original wave, there would be no difference. And then neither would there be any difference in the energy of the electron that was knocked off the metal. But scientists found to their great surprise that only if they turned up the frequency of the light was the energy level of the electrons affected. Instead, if they turned up the intensity the number of electrons which where knocked off raised.
The Solution
This one can be understood by thinking of light as be made up be photons. The energy of a photon depends only on its frequency(as stated in the black body radiation) and when you turn up the frequency of the light beam you turn up the frequency of the photons in it there by giving them higher energy, and the higher energy they have the more energy can be given to the electrons that they hit, making them more energetic. But if you turn up the intensity of the light beam you turn up the number of photons that is emitted, so a higher amount of photons will hit a higher amount of electrons thereby creating a larger number of electrons being knocked of from the metal.