Why is Maxwell's theory insufficient to explain the photoelectric effect? You can argue that electromagnetic waves from a UV light source travel towards a metallic plate,  and by the time they reach where a loose electron is located, they affect it with a electromagnetic field (force), so the energy might be enough to knock off the loose electrons. So why is a photon hypothesis proposed?
 A: Photoelectric effect can be observed instantaneously when light is flashed and it is independent of intensity of the light. If wave model were at play here, intensity of the light (amplitude of the wave) would have an effect on how quickly an electron receives sufficient energy to be knocked out. 
This is because energy in a wave is related to its amplitude. 
A: Further to the above answers, only the particle model of light can explain one key observation: if the frequency is too low no intensity is sufficient to strip away electrons, whereas even a low-intensity high-frequency light beam can do it. That the amount by which such frequencies are excess is proportional to the voltage generated, i.e. the kinetic energy of the electrons, is telling too. But the clincher is that, provided a current (i.e. electrons per second) can be generated, it's proportional to the intensity-to-frequency ratio (if intensity is measured as power per area). The interpretation is immediate: one photon, with energy proportional to the frequency, frees a single electron.
A: Maxwell Equation do not take into account the quantum nature that comes after introducing the Planks constant. The energy of a photon is Planck constant X frequency. There is no Planck constant in Maxwell equations in this sense they are termed classical. To explain photoelectric effect the quantum nature of light has to be assumed.
A: Because the (kinetic) energy of the emitted electron is proportional to the frequency of the light, not the intensity (a measure of amplitude). 
Imagine a beach ball sitting on the shore of the ocean. Compare two scenarios:
1) Very large waves, which do not come very often, strike the beach ball (high intensity, low frequency).
2) Small waves, one after the other in quick succession, strike the beach ball (low intensity, high frequency). In which case will the beach ball be sent flying faster in the air (ejected with higher kinetic energy)? 
One might expect intuitively that a very large wave would sent the beach ball flying faster than... what? One small wave? Does the fact that many more small waves follow the first small wave even matter, since the first wave will send the beach ball flying away (at a low speed)? 
Well, it turns out experimentally that the kinetic energy of the ejected electron (speed of beach ball) depends on frequency of the waves, not the intensity. That is, a bunch of small waves will send a beach ball flying faster that a few big waves. But only the initial wave can really strike the beach ball, because after that it is sent flying away. So it is not "logical" for a small initial wave to give the beach ball more speed than a big initial wave... Unless we rethink what the "waves" are. 
Unlike our beach analogy, where the water surely travels in waves, we are not so sure that light must strictly be a wave (because we cannot observe its structure with our eyes). Einstein won his Nobel prize showing that if light is thought of as a particle, whose energy is proportional to the associated frequency (E=hf), then the photoelectric effect is fully explained (Einstein's Nobel Prize helped lay the foundation for quantum mechanics, and had nothing to do with General and Special Relativity Theories for which he became so famous). 
This is the quantization of light, and since it is quite non-classical, it would be hard to interpret it in terms of our beach metaphor. I suppose you could say that based on experimental data, we conclude the "ocean" (as it behaves in this experimental setup) must not be composed of waves, but rather arrive all in one packet. That is, the low waves of rapid succession are in fact a single packet whose energy is quite large because it is proportional to the rapid frequency; thus it sends the beach ball flying with great speed. The naively expected case, waves of great height that do not come very often, are re-interpreted as a packet of low energy (due to the low frequency), and thus do not send the ball flying with very much speed.
So the idea that light is quantized as photons explained the unexpected data from the photoelectric effect experimental setup. Does this mean light is strictly composed of particles? No. It means light behaves as particles in this specific experiment. Other specific experiments show that light behaves as a wave in those cases. In the end, we must conclude that light is its own beast, which can sometimes be described as a particle, and sometimes be described as a wave. We call this idea Wave-Particle Duality. In the end, light is not necessarily either, the wave picture and particle picture are merely models to help us rationalize how light behaves in different situations.
