First of all I am a novice regarding my knowledge of quantum mechanics. But curiously I do want to know what is the problem if energy is continuous like spontaneously flowing tap water.
In fact I actually don't know what is continuous referring here. What does this statement mean? Energy is the ability to do work. So, what is the problem with continuous energy?
Plank solved the problem of Classical physics by chopping up the energy into discrete particles which possess energy proportional to the frequency of the radiation.
So, what is this? What led Planck to chop the energy? What is the physical intuition behind this? Please help giving me a math-free explanation.
Answer
Let me first give you an example on what 'continuous' and 'discrete' are, and then show you how it relates to energy.
Let's say water is flowing in a stream. We say that the water flow is 'continuous' since we don't see induvidual 'blocks' or 'lumps' of water flowing one after the other in the stream. All we see is one continuous indivisible 'body' of water which negotiates rocks and ridges with ease by simply 'flowing' over them.
It may seem apparent at first that energy also behaves in the same manner. The way we think of energy untuitively is that it is some sort of invisible continuous fluid which floats in space from body to body. The evidence to support this intuition physically is not scanty.
Look at the example of visible spectra. This is what you get when you pass a beam of white light through a prism. Prior to the 20th century, the theory that explained the properties of light satisfactorily was the Wave theory of light. According to it, each color of light corresponds to a particular wavelength, and thus also corresponds to a particular energy with which it moves. Since this spectra is continuous, it means that the white light has all energy levels or colors, there are no gaps or sudden color changes. Can you make out exactly where, for example, yellow turns orange ? 
But then take a look at this spectra too: 
Do you see bold, discrete and distinct lines of red and blue ? This is a hydrogen spectra. The colors are the different wavelengths of light emitted by hydrogen gas filled in a tube when current is passed through it. These distinct lines baffled 19th century physicists. They couldn't understand how only particular 'discrete' wavelengths of light could be emitted by hydrogen atoms and not 'continuously' like the visible spectra.
At the start of the 20th century, Max Planck was conducting an experiment concerning black body radiation. By chance, he observed that in the energy values he tabulated, the values were always integral multiples of $h\nu$, where $\nu$ is the frequency of radiation emitted by the black body. It conlusively meant that energy could only be emitted in specified amounts and had an elementary, basic unit much like the 'lumps' of water I described in the second paragraph. If energy is transmitted in space from body to body, it means that that energy is made up of such elementary 'lumps' called 'quanta' (later they termed it as 'photon', a particle-like entity), much like all mass is made up of 'atoms'.
The 'discrete' wavelengths in the hydrogen spectra was due to the atoms collectively emitting 'discrete' photons of specified energy and not of any energy (as in the continuous spectrum) as electrons jumped to lower energy levels from higher ones.
Another question may arise: 'Why' does energy have to only be transmitted in discrete, specified quanta ? The answer ultimately is that it 'just is'. Later in the 20th century, Einstein showed that energy and mass are two sides of the same coin. So one may also argue that: Just as mass has a quanta called 'atom', energy also may have such a quanta in the form of 'photons'. Eitherwise, none of these can be given as 'the one and only reason' since it is purely a question of philosophy, not physics.
P.S: As julian fernandez said, it will take a really long time to fully type out the origins of quantum mechanics. What I have given is just a brief intro to the field.
Hope it helps !
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