While the wavelength/frequency of the incoming light does affect the results of the experiment, the key is that there is a certain frequency/wavelength beyond which no current is detected, no matter the intensity of the incoming light.* This is odd because if light behaved like classical waves you should be able to simply increase the intensity of the light without changing the frequency and get ejected electrons, and the energy of each ejected electron should increase as the intensity of the light increases. But this doesn't happen. The simplest explanation that matches all observations is that light interacts with matter discretely, not continuously. By that I mean that energy from the light wave isn't absorbed in a continuous manner, as a classical wave would behave, but in discrete "packets", where each packet of energy is absorbed all at once. The amount of energy in each packet increase as frequency increases, explaining why the energy of the ejected electrons increases with the increasing frequency of the light. These packets of energy are called photons.
*Note that once the frequency of the light is high enough to eject electrons, the intensity of the incoming light can be reduced to nearly zero and you still get ejected electrons with the same energy as when the light intensity is much higher, albeit at a very low rate. This should not happen if light were just a classical wave and is further evidence that light has a particle-like behavior. If the light wave behaved classically, the electrons would absorb energy continuously over time, which would mean the electrons are behaving more like they do in a radio antenna. In an antenna, the electrons absorb energy in a near-continuous manner (owing to the very low energy per photon or radio waves) and this energy is dissipated as heat. No electrons are ejected from the antenna unless the intensity of the incoming radio waves is extremely high. So you can think of it as needing to deposit energy into the electrons faster than they can get rid of it as heat.
The photoelectric effect shows us that this behavior is only an approximation of a classical wave at low frequencies and that it fails when the frequency of the light is high enough. Classically, the incoming light would not eject any electrons below a certain intensity and increasing the intensity would always allow us to eject electrons. Instead, we get electrons regardless of intensity as long as the frequency is high enough and increasing the intensity of the light has no effect if the frequency is too low. As I said before, the best explanation is that light is not purely a classical wave and consists of photons.
