The sorcerer, as we know it, dates back to D&D 3e, which was the first under Wizards of the Coast. To date, to my knowledge, they have never gone into a lot of detail about this kind of thing.
In 3e, even the whole concept of sorcery coming from a bloodline wasn’t really a part of the original description.
Some sorcerers claim that the blood of dragons courses through their veins. That claim may even be true in some cases—it is common knowledge that certain powerful dragons can take humanoid form and even have humanoid lovers, and it’s difficult to prove that a given sorcerer does not have a dragon ancestor.
(D&D 3.5e Player’s Handbook pg. 51)
The “bloodline” concept—and in 3.5e it was solely and explicitly draconic bloodlines—got solidified more during 3e and the “3.5 revised edition,” but other than “well sorcerers do it the same way dragons do it,” which just shifts the question rather than answering it, we don’t get more detail on how sorcery actually works. There were a lot of books about dragons (Draconomicon, Races of the Dragon, Dragon Magic, Dragons of books for each setting, etc.), which describe some of this, but most of it is like this:
In essence, dragons are created from a blend of magic and the most basic elements, bonded powerfully to the magical world. Just as an individual magic item or locale might be a focal point for magical energies, dragons are focal points for the flow of magic—living, breathing conduits between its raw potency and the rest of the world.
(Dragon Magic, 3.5e, pg. 59)
It doesn’t really tell us a whole lot, and that doesn’t even begin to cover non-draconic sorcerers, since those weren’t even really a thing in that edition.
The sorcerer in 4e—where it was not a core class, but introduced in Player’s Handbook 2—got to choose a “Spell Source,” with “Cosmic,” “Storm,” and “Wild” joining “Dragon” as options. In general, 4e was a bit... fluff-light, we’ll say. The sorcerer gets a couple sentences each about their role and power source, and then three short paragraphs of description, and that’s it.
Anyway, on power source, this is what we’ve got; note that Player’s Handbook 2 only had Dragon Magic and Wild Magic, and Cosmic and Storm would come later.
Arcane magic is in your blood, as a touch of either ancient draconic power or untamed chaos energy, and you unleash it through sheer force of will and physical discipline.
(Player’s Handbook 2, 4e, pg. 136)
And then in 5th edition, well, you’ve already quoted the most relevant details we have.
So there just isn’t much more than “it’s in their blood” to be had. Given that, I’d like to offer some, say, informed speculation. None of this is official, but I’m pretty sure none of it is “officially wrong,” either, that is, I’m pretty confident nothing out there is going to contradict me.
Dungeons & Dragons started out in 1974 with just three classes: fighting-man, magic-user, and cleric. The magic-user would be the one that matches what we would call the wizard today. Nothing like the sorcerer existed, or would exist until 2000’s 3rd edition Player’s Handbook. The wizard, then, is the original “arcanist,” and indeed, it most closely matches the characters in the novels by Jack Vance, whence we get the term “Vancian” to describe D&D’s magic.
Thanks to the Jack Vance’s novels, plus the 26 years of D&D that preceded the existence of the sorcerer, we know a bit more about how wizardry actually works. It’s not super clear that all the old lore—to say nothing of the novels that inspired Gygax but were fundamentally independent—actually still applies “canonically” today, but again, nothing out there outright contradicts these ideas. Wizards study arcane patterns, formulae, and the like. They have to memorize with utter precision the weft and weave of the arcana involved in each spell, the pattern, the gestures, the utterances, and so on. In Vancian spellcasting, preparing the spell is where most of the work actually takes place, when you can dedicate time and care, and reference your materials. You build up the spell in your mind, and store it there, just incomplete, so that you can finish it in a moment of “casting” it. The prepared spell is an actual thing that exists in some mental dimension, made out of magic and living in the wizard’s brain.
(For those players only familiar with wizards of 4e and/or 5e, I should note that prior to 4e, wizards had to prepare not just which spells they wanted to use, but also how many of each. Each spell slot was spoken for, and if you had five 1st-level spell slots, you might prepare three magic missile spells and two grease spells—and if you ended up needing a third grease spell, you were out of luck. This is closer to how things work in Vance’s novels. The changes to how preparation work in 4e and 5e—which are not at all similar in this regard—probably don’t really change anything on a fluff level, since the prepared spell just needs to be somewhat more “durable” to allow its re-use so long as you have spell slots, but I thought it might help with the imagery to know that originally each spell was prepared separately.)
What does this mean for the sorcerer? Well, their “casting” of a spell is just like a wizard’s—they use the same gestures, the same utterances, and so on. A wizard can recognize the spell that a sorcerer is casting and vice versa (explicitly, in 3e; 4e and 5e get a little more nebulous on this in my experience). So the sorcerer must also have a “prepared spell” somewhere, only they didn’t prepare anything. But since the prepared spell is an actual, physical thing that the wizard created out of magic, all we need for the sorcerer to have one as well is just to imagine that someone or something else created that magical structure. It lives within the sorcerer just the same way that the wizard’s does, only it’s permanent and the sorcerer didn’t have to make it.
So what did? I like to imagine it was deoxyribonucleic acid (DNA). I’m serious—why not? DNA and RNA and the associated mechanisms are responsible for our cells producing exceptionally complicated molecular structures—some of the proteins our bodies make, using the blueprint found in DNA, are preposterously complicated. If magic lives in everything in the world of D&D, then it can live in cells—and the mechanisms of the cell can potentially manipulate it just as they manipulate chemicals. Perhaps, in a magical world, that’s even necessary, that all creatures necessarily create those little bits of magic that enable their functioning.
And sorcerers? They’ve just got genes for actual spells in their DNA. Inherited from dragons, or mutated by the whims of Chaos, or whatever. It doesn’t, obviously, have to actually be DNA, it’s just some genetic part of the sorcerer’s body that creates these magical structures that a wizard has trained himself to do manually (well, mentally). An autonomous process, so much easier, but a sorcerer maybe doesn’t actually get to decide what spells they have—maybe they just grow into using more of the spells that were already encoded in their DNA.