Classical Wave theory and the photoelectric effect

Electrons ejected from a sodium metal surface were measured as an electric current. Finding the opposing voltage it took to stop all the electrons gave a measure of the maximum kinetic energy of the electrons in electron volts.

What does intensity "continuously variable" mean?

It means that for any frequency one can make a continuous increase in the energy of the beam of light.

How does the intensity of light being continuously variable make it unable to explain the photoelectric effect?

As is seen in this particular experiment no electrons come out bellow a certain frequency even though the material is irradiated with it. This means that the photo electron energy cannot be built up by the intensity of the light shone on the material.

Measurements indicate that in the photoelectric effect, the energy of the electrons which leave the surface depends on the frequency of the incoming light, but not on the intensity of the light. This indicates that the incoming light consist of bundles (or quanta) of energy whose size is determined by the frequency. The number of electrons emitted depends on the intensity of the light (or the density of quanta.)