In any metal, there are free surface electrons. For these electrons to escape from the surface of a metal, they require a specific amount of energy, called the 'Work Function'. There are many sources of this energy, but we will focus on just one - electromagnetic energy. Traditionally, EM energy was considered as a transverse wave, similar to that of water waves, but in the Quantum model, the energy is split into small packets, called photons.
These photons behave similarly to particles, and when they come into contact with a metal surface, they interact with only one surface electron. They transfer all of their energy to that one electron, and if this energy is larger than the Work Function (as mentioned previously) the electron will escape the metal as a 'photoelectron' (hence photoelectric effect). Since only one photon can interact with each electron, a greater number of photons incident on the metal per second has no effect. An increase in the EM photon energy means that the released electrons have more energy left over after being emitted, in the form of Kinetic energy.