With the photoelectric effect they found that only light above a specific frequency (threshold frequency) would eject electrons from the metal, regardless of the intensity of the light or the time the metal was exposed.
The wave model predicted that if the metal was exposed to light for a long period of time, the electrons would eventually accumulate enough energy to escape from the metal, however this was not the case. They also predicted that a higher intensity would 'make up for' the lower frequency, and electrons would still be ejected, but again this was not found.
Only was to explain it is through photons/quantum ideas. Electrons can't gain energy incrementally, they either absorb a whole photon of hight enough energy and are ejected, or don't absorb one at all. So they are only ejected when the energy of the photons (frequency of the light) is high enough. The intensity corresponds to the number of photons.
Hope that helped.