Einstein used the idea that light consists of a small package of energy called photons to explain the photoelectric effect. Since a beam of light consists of an enormous number of photons, intensity of light (brightness) is related to the number of photons, but not to the energy of each.
Since there are a vast number of photons in a beam of light, the intensity of light (brightness) is correlated with the number of photons, but not with their individual energies. Therefore, to release an electron from the surface, a photon with a specific minimum energy should be absorbed. Since energy and frequency are inversely correlated (E = hf), the model says a threshold frequency that correlates to the lowest energy (metal's work function) needed to release the electrons.
There really is no time lag since electrons release as soon as they take up enough energy from photons. There is no lag time since a current will start flowing as quickly as a photon with adequate energy enters the metal plate.
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