A little more should be said. The fine structure constant isn't just a figment of someone's imagination.
But first, listen to Feynman. "You can understand relativity, see after a guy proposed it why it must be so. But, nobody understands quantum mechanics."
Second, there is something called a "black body"--a perfect absorber and perfect emitter of radiation (the sun comes close). So, people worked up a theory that would calculate the energy in a black body using what are called "harmonic oscillators"--basically little springs with small masses attached--int the walls. When they did the calculation, the energy went to infinity. This was called the "ultraviolet catastrophe" (physics is dramatic). Max Planck (you certainly know of him in Deutschland) solved the problem. Instead of using an integral (a calculus concept) which allowed each oscillator to have any energy, he assumed that only certain energies were allowed for each "frequency" v (v, 2v, 3v, etc.) and did an infinite sum. That solved the problem. No one understood why it should be so, but the little packets of energy corresponding to integers were called "quanta". This was in 1905 or so.
Meanwhile, people had been shining light on hydrogen to see what would be "absorbed" and would would come out. It turned out that only certain "wavelengths" (colors) were absorbed or emitted--the energy was also "quantized".
Why? A long process followed, spearheaded by Heisenberg and Schoerdinger and an equation was developed (for a single H atom it can be solved, but to this day the meaning of the solution is disputed--the absolute square of the solution at each point in space around the proton is said to be the probability of finding the electron in that place (and it can be 0--no chance at all). This was around 1926. The solutions could also lead to the energy of the electron, and it was found that electrons around the hydrogen's proton (in "orbits") could only have certain energies. Again, they were "quantized". This explained the light spectra--an electron, depending on which orbit it was in, could only absorb a single color of light to move to a higher orbit, and when "tickled" to move to a lower energy orbit could only emit a specific color. (These spectra--the colors a given atom can emit--are so characteristic of the element that they can be used to determine the composition of stars based on the starlight, and were used to identify new elements before the Periodic Table was filled in. There are lots of complications because atoms and stars move and there are so many of them, but nonetheless it all works.)
So, where does the constant come in? Schroedinger's equation wasn't perfect. There were some faint lines close to the bright ones that weren't accounted for, the "fine structure". Your man Pauli proposed that electrons had "spin" (like a little round top). This helped, but it took an equation that was consistent with relativity (Schroedinger's is not--Dirac finally came up with one after others had failed) to explain them. Finally, the "fine structure" was explained, and the constant appeared in the results.
The math for all this is complicated (but there is math that is far more complicated) but the real problem is that one can't really see where the equations come from ("see why it must be so"). People just bumbled into them--it is not clear, at least to me, how they thought them up.
You recall the "Higgs boson"? ("Bosons" and Fermions" come from Pauli's spin--fermions like electrons in the same "state" cannot occupy exactly the same space at the same time because they have "spin"--the spins at least must be different, but "bosons" without spin bunch up: this is called the "Pauli exclusion principle".) The math is getting bad, and Higgs himself was asked about his discovery and the modern explanation of it. "I should have taken more maths in school--I can't really follow what they are doing anymore."
Oh, and there are other things nobody really understands (even though they can be calculated to a good approximation). For instance, Dirac's solution calls for a "quantum vacuum"--particle pairs periodically just pop out from nowhere in empty space and then "annihilate" each other. (This is the basis also of the postulated "Hawking radition" that Hawking proposed is emitted from black holes--which aren't supposed to let anything out. The idea is that if a particle pair "pop out" around a black hole, and the hole eats one of the pair, then it has to emit some energy to compensate so the energy of the universe doesn't change. There is something called the Casimir force based on these phantom pairs which has experimental support, so it's not just a fantastical theory.)
Also there is also what Einstein called "spooky action at a distance" which is analogous to two twins on opposite sides of the world. If one drops dead, then the other must also at the same instant. This was shown by the late John Bell (who like Hawking died before he could get a Nobel Prize because the experiments hadn't been done to confirm it yet).
"Nobody understands quantum mechanics". Feynman also asked whether the universe was like an onion. You peel off one mystery by explaining it, and a new mystery layer is exposed. Well, it gives people something to do.
