My favorite deep, elegant and beautiful explanation is Albert Einstein's 1905 proposal that light consists of energy quanta, today called photons. Actually, it is little known, even among physicists, but extremely interesting how Einstein came to this position. It is often said that Einstein invented the concept to explain the photoelectric effect. Certainly, that is part of Einstein's 1905 publication, but only towards its end. The idea itself is much deeper, more elegant and, yes, more beautiful.
Imagine a closed container whose walls are at some temperature. The walls are glowing, and as they emit radiation, they also absorb radiation. After some time, there will be some sort of equilibrium distribution of radiation inside the container. This was already well known before Einstein. Max Planck had introduced the idea of quantization that explained the energy distribution of the radiation inside such a volume. Einstein went a step further. He studied how orderly the radiation is distributed inside such a container. For physicists, entropy is a measure of disorder.
To consider a simple example, it is much more probable that books, notes, pencils, photos, pens etc. are cluttered all over my desk than that they are well ordered forming a beautiful stack. Or, if we consider a million atoms inside a container, it is much more probable that they are more or less equally distributed all over the volume of the container than that they are all collected in one corner. In both cases, the first state is less orderly: when the atoms fill a larger volume they have a higher entropy than the second one mentioned. The Austrian physicist Ludwig Boltzmann had shown that the entropy of a system is a measure of how probable its state is.
Einstein then realized in his 1905 paper that the entropy of radiation (including light) changes in the same mathematical way with the volume as for atoms. In both cases, the entropy increases with the logarithm of that volume. For Einstein this could not just be a coincidence. Since we can understand the entropy of the gas because it consists of atoms, the radiation consists also of particles that he calls energy quanta.
Einstein immediately applied his idea for example to his well-known application of the photoelectric effect. But he also realizes very soon a fundamental conflict of the idea of energy quanta with the well-studied and observed phenomenon of interference.
The problem is simply how to understand the two-slit interference pattern. This is the phenomenon that, according to Richard Feynman, contains "the only mystery" of quantum physics. The challenge is very simple. When we have both slits open, we obtain bright and dark stripes on an observation screen, the interference fringes. When we have only one slit open, we get no stripes, no fringes, but a broad distribution of particles. This can easily be understood on the basis of the wave picture. Through each of the two slits, a wave passes, and they extinguish each other at some places of the observation screen and at others, they enforce each other. That way, we obtain dark and bright fringes.
But what to expect if the intensity is so low that only one particle at a time passes through the apparatus? Following Einstein's realist position, it would be natural to assume that the particle has to pass through either slit. We can still do the experiment by putting a photographic plate at the observation screen and sending many photons in, one at a time. After a long enough time, we look at the photographic plate. According to Einstein, if the particle passes through either slit, no fringes should appear, because, simply speaking, how should the individual particle know whether the other slit, the one it does not pass through, is open or not. This was indeed Einstein's opinion, and he suggested that the fringes only appear if many particles go through at the same time, and somehow interact with each other such that they make up the interference pattern.
Today, we know that the pattern even arises if we have such low intensities that only one, say, photon per second passes through the whole apparatus. If we wait long enough and look at the distribution of all of them, we get the interference pattern. The modern explanation is that the interference pattern only arises if there is no information present anywhere in the Universe through which slit the particle passes. But even as Einstein was wrong here, his idea of the energy quanta of light, today called photons pointed far into the future.
In a letter to his friend Habicht in the same year of 1905, the miraculous year where he also wrote his Special Theory of Relativity, he called the paper proposing particles of light "revolutionary". As far as is known, this was the only work of his that he ever called revolutionary. And therefore it is quite fitting that the Nobel Prize was given to him for the discovery of particles of light. This was the Nobel Prize of 1921. That the situation was not as clear a few years before is witnessed by a famous letter signed by Planck, Nernst, Rubens and Warburg, suggesting Einstein for membership in the Prussian Academy of Sciences in 1913. They wrote: "the fact that he (Einstein) occasionally went too far should not be held too strongly against him. Not even in the exact natural sciences can there be progress without occasional speculation." Einstein's deep, elegant and beautiful explanation of the entropy of radiation, proposing particles of light in 1905, is a strong case in point for the usefulness of occasional speculation.