What does the photograph show? Well, it shows a sentence written by Dirac himself.
In 1956, he bought an air ticket to Moscow. The main purpose of his trip was to write the important sentence on a blackboard – well, it was the blackboard of Russian physicist Dmitry Ivanenko (1904-1994). On October 3rd, 1956, Dirac finally wrote this important sentence. Ivanenko and his comrades appreciated the importance of the sentence so you can still see the quote on that blackboard. If a janitor mistakenly erased it, Vladimir Putin would rightfully send the janitor to a Gulag. ;-)
It's not hard to see why Dirac is so naturally associated with the concept of beauty in theoretical physics. His contributions to physics have been beautiful. The derivation of the Fermi-Dirac statistics from the anticommuting fields is beautiful. The bra-ket formalism of quantum mechanics is beautiful. Most obviously, the Dirac equation\[
(i\gamma^\mu \partial_\mu - m) \Psi = 0
\] is beautiful. This quantum mechanical equation describing a spinning relativistic fermion generalized the previous equations – Schrödinger's or Pauli's equation – in a way that respects the Lorentz covariance and some other (e.g. discrete) symmetries, that can be expressed by a small number of characters (especially if you use the "slashed" notation for contractions with the gamma matrices), and whose very existence was surprising.
Also, we may say that the equation is beautiful in the same sense as a femme fatale: a bright enough man is instantly charmed when he sees her or it for the first time. He wonders how he could have lived without knowing her or it. He wants to remember her or its shape and curves because they seem to be perfect, they seem to fit together, there exists no simple enough (or sufficiently nearby) way to deform or adjust it that could be considered an improvement. Any modification would make her or it worse.
These are some informal reasons why the Dirac equation is pretty. Now, for a contemporary theoretical physicist, the Dirac equation is a rather trivial thing so its "fundamental beauty" is being exaggerated. But the principle that beauty matters holds more generally. Even when one considers more modern and less trivial insights in theoretical physics than the Dirac equation (such as grand unification, supersymmetry, string theory as a whole, and lots of individual aspects of string theory), it's still true that the important ones are true in an analogous sense as the Dirac equation.
Now, Sabine Hossenfelder wrote another rant against theoretical physics, against string theory, and especially against beauty in physics. Her main thesis is:
Nature has no obligation to be pretty, that much is sure.Oh, really? Is it "sure" that Nature has no obligation to be pretty? A much more sure proposition is that her sentence above directly contradicts Dirac's quote on the Moscow State University blackboard that was reproduced in the very title. According to Dirac and many others, Nature actually does have the obligation to be pretty. This statement wasn't obviously true a priori. But there exist lots of evidence that the statement actually is true a posteriori.
Beauty and the truth aren't synonymous but there exist totally rational reasons why a prettier candidate for the law of physics is more likely to be true than an uglier candidate. The beauty in the Yes/No sense – i.e. a qualitative property that "a femme fatale optimizes a certain quantity that measures her beauty quantitatively" – divides theories to two groups and the right way to do Bayesian reasoning is to assign comparable prior probabilities to qualitatively different hypotheses. In this sense, a "pretty" theory is as likely a priori as an ugly one, but because there are so many ways in which one may be ugly, the pretty theory ends up being special and therefore dramatically overrepresented according to the probabilistic distribution.
If you observe that a candidate theory has some symmetry or any other "special feature" that looks like an aspect of beauty, the same principle is still true. One can't turn any of these arguments into high-precision science (at least today, we don't know how to do it) but the general vague argument has both rational reasons to be right – as well as lots of circumstantial, basically empirical evidence that it is actually true.
You can reformulate the same argument with some theological flavor, too. When God was creating the Universe, He needed to pick some laws of physics. He probably didn't do that job too hastily. Instead, He probably wanted good enough laws – or the best laws according to some job tender. The beauty of the proposed laws is at least a natural virtue that God could have preferred.
Now, Hossenfelder doesn't find anything strange about her cheap negation of Dirac's important quote. Her "good friends" in quotation marks are shocked, we're told. Surely she doesn't mean it, they say. But she does. She understands literally nothing about the important lessons of 20th century physics, she has no innate aptitude (let alone intuition) for theoretical physics whatsoever, but the political correctness that has run amok has encouraged worthless individuals from privileged groups such as herself to simply trash Paul Dirac, his quote, his equation, all other important insights in modern theoretical physics – and pretend to be important at the same time.
I am actually annoyed by the politically correct "good friends" (including some string theorists who have worked at the places where this fake physicist has pretended to work) who constantly nurture the illusion that she can't mean what she says. She can't be this dumb, this dishonest, this ugly, this disgusting, and so on, can she? Well, she surely is. If you weren't deliberately fooling yourself all the time, "good friends" of hers, you would see it almost immediately. If you hybridize some of the most deluded, dishonest, disgusting, and stinky building blocks and excrements from some of the most notorious jerks and professional crackpots – including Mr W*it and Mr Sm*lin – and if you mix these things with some extra distasteful stuff, what you end up with is Sabine Hossenfelder.
The physical laws should have mathematical beauty – and their vitriolic foes should be disgusting, lying, untalented likes of Sabine Hossenfelder. That's how the roles are divided in our Universe.
And that's the memo.