Why did people expect the number of electrons emitted because of the photoelectric effect to rise with the radiation's intensity?

The number of electrons emitted does, in fact, increase with the intensity of the radiation. Higher intensity means more photons, which means more chances to knock an electron loose.

It's the energy of the emitted electrons that people expected to increase that doesn't. In a classical wave picture, an increased intensity means a higher amplitude of the wave, which means more dramatic "shaking" of the electrons, and should thus lead to electrons being ejected with higher energy. In the quantum model, the emission is caused by a single photon, and the energy is determined by the frequency of that photon. Thus, the energy depends on the frequency but not the intensity.