What did Grothendieck mean by "the capacity to be alone" in the context of mathematical research?
As this is a translation, the words "capacity to be alone" may not be the best way to phrase it. Perhaps this is a more literal translation, but from context I might write it as "independence".
He is saying that other people who he perceived as more capable individuals ("more brilliant than" he) as researchers were successful, and yet were constrained by doing work similar to others. Perhaps they worked on incremental solutions to problems within a general framework in the research community.
He implies that the status quo ("invisible and despotic circles which delimit the universe of a certain milieu") is a constraint on creative work.
He is suggesting that his own impact has been more profound or more revolutionary because he has been able to be more independent from others, to make his own path either in the problems he approached or in the solutions he found, and that he thinks the other people he talks about would have had similarly profound impacts if they had been less focused on the work or opinions of others.
Who knows whether he is correct that this is how researchers should think, it's just one opinion. I think it's somewhat dangerous to assume that because an approach worked for someone that you should take their advice that it would work for everyone. There is a lot of survivorship bias in that those who go their own path but are not successful are seldom heard from - who would read their book?
I'm not a mathematician and am not familiar with Grothendieck's work. Just from his Wikipedia page it seems he became a bit reclusive and didn't engage with the broader community later in his career. From Wikipedia which talks about the work that you quote from:
In the 1,000-page autobiographical manuscript Récoltes et semailles (1986) Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner but which he progressively perceived to be governed by competition and status
It could be that he misjudges the ways in which his earlier work that became so influential benefitted from that of others around him, and misjudges the profundity or impactfulness of the work that he later pursued in solitude. Just because someone has a genius in mathematics or any other field does not necessarily mean they are good at assessing the reasons behind their own success; self-reflection is an entirely separate skill set.
For what it's worth in contrast, another famous mathematician wrote, "If I have seen further it is by standing on the shoulders of Giants", which seems to be a bit of a different philosophy (though I may be treating Grothendieck uncharitably, as it seems his quarrel is less with the work of others and more with the academic establishment, but it still speaks to the value or lack thereof of independence/solitude in academic work). Google Scholar uses the motto "Stand on the shoulders of giants" in recognition and tribute to this sentiment.
All indications are that Grothendieck did not lack self-confidence. Just as well, considering his harsh and unlucky childhood and life situation prior to academic mathematics success.
It's not really that he "created a new subject" (modern algebraic geometry), but, rather, apparently found a path (that others were also seeking) to revivify the subject. He did have J.-P. Serre's support by correspondence, which not all of us do. :) Still, the homological ideas in his Tohoku J. article (in particular emphasizing sheaf cohomology as right-derived functors of the global-section functor) were not "entirely new", as Eilenberg and H. Cartan write their book at about the same time. As far as I know, though, the sheaf-cohomology idea was new, with Cech cohomology still a competitor into the 1960s.
A. Weil's 1949 "Foundations of Algebraic Geometry" was the main preceding attempt to rigorize (and extend...) the intuitive physical geometry of the Italian school from early 20th century. It did not seem to offer a good extension to deal with the Weil Conjectures, which Grothendieck had set his sights on.
He was evidently a forceful personality (I never met him) as well as a very good mathematician, and managed to persuade a small army of collaborators to work to implement/develop his vision.
At some point, evidently, he did become disenchanted with French academic culture... and "retired". To my mind, there is a considerable irony that the "establishment" he decided to disdain nevertheless supported him... and he apparently accepted most of that support.
Coming back to the specific question: I think it really amounts to a difference of taste. Grothendieck could not help himself but be rebellious, and disdained those who were not rebels. Oop, not counting the people who worked very hard to make his visions come to life? I myself am unsympathetic to this, but nevermind.
So, if we are to take it seriously, his claim was that those other talented people could have been as "great" as he, Grothendieck, if only they would have adopted his approach, instead of ... following their own?
At best, it seems to me that his remarks confound the institutions with the people.
EDIT: hoping to clarify, in response to some comments... First, I do not at all wish to diminish AG's rightfully earned reputation as one of the luminaries of the 20th century. But/and I would want to say that much of his legacy was, in fact, a team effort, with him as a/the leader. But, also, for sure, several or many other exceptional people "helped"! :) This fact is not as often made clear in the heroic legends. AG can be a heroic figure without diminishing the other very, very good mathematicians who made huge contributions (if not quite in his style).
Although I myself may be a small-time iconoclast, one thing I've thought a lot more about in the last 10-20 years is to try to give credit where credit is due. And not just to the "top ten" or similar most-heroic figures. And, factually, as my own scholariship has improved, I've learned that lots of great ideas were already manifest decades or a century or two earlier... but somehow not necessarily achieving top billing in textbooks and other professional-cultural mythology.
In AG's case, I think the Serre-Grothendieck correspondence shows the scholarly inputs Serre provided AG, although it is not easy to see that reflected in EGA nor SGA. The style of the latter did not seem much concerned with bibliographic/historical completeness.
I do think that one part of the Bourbaki impulse was an instance of "shedding the past"... partly for good reasons, but, also, throwing some babies out with the bathwater.
Many of the responses here give correct answers. They also explain that the view of this quote is not the norm, and that one shouldn't necessarily trust Grothendieck's advice based anyway. I agree with all of this, BUT with caveats so significant I believe it deserves a separate response.
Caveat 1: He is not implying that one should reject other people in the field and not cooperate with them.
As a matter of fact, in the same text he talks about how almost all his first ideas in algebraic geometry were stimulated by Serre. He was certainly no stranger to collaboration, and this is completely compatible with this quote. Rather than explaining abstractly this compatibility abstractly, it may be more explain to learn how he allegedly worked. In his IHES days, he would present his seminar once a week and follow it up with discussions. Sometimes he would also often invite others to his home to do math. But on other days, he would work alone, often late into the night, to come up with the notions he would discuss with other mathematicians.
The most well-known example of this is the notion of a scheme. This idea, or at least close variants of it, were up in the air at the time. But how did Grothendieck know that the very foundations of algebraic geometry should be rewritten with them? (a truly gargantuan task) According to this same text, he had to work alone, plumbing this concept himself to see this was the way to do it all. Another concept, the "Grothendieck topology" generalized the very notion of a topology in a seemingly naive way. But this was what was necessary to define etale cohomology, which led to the proof of the Weil conjectures.
For a more detailed discussion, see here. It seems to me that whenever he did math, he had a grand vision. This vision was certainly stimulated and enhanced by others, but it was something he could "see" on his own that was remarkably unconstrained.
Caveat 2: This quote is less a judgement of others than an expression of something personally important.
While this quote does indeed mention the impact of other mathematicians, I don't think that is the point of the paragraph. If you read on, you will find that this section is about the "interior adventure", especially about his experience of mathematics. He explains that for him, innocence and his ability to listen to the nature of things were the most essential traits to his success in math. While this formulation may sound romanticized, the content seems sound to me.
The notion that other mathematicians would have gained from what he describes is, in my opinion, a valid one. I believe most of the pushback against it comes from the fact that he is saying about himself, which may lead some people to associate it with arrogance. As a matter of fact, I have heard people talk about how various fields could have used a Grothendieck without any controversy.
Caveat 3: Just as one should not accept advice just because it is from Grothendieck, one should not reject it just because it is from Grothendieck.
There are some rather disparaging comments in the other answers about Grothendieck as a person, and how we should not listen to his advice as a result. I find this attitude very troubling for two reasons.
First, it is largely based on negative stereotypes that are absolutely false in this case. I've even seen many of the concrete facts about him be twisted or even made up to promote this sensationalist archetype of a recluse mathematician who understands nothing but math. Having read some of his later writings, I can confirm that such a depiction of Grothendieck is completely wrong. For sure, some of the events in his life did superficially resemble the stereotype, but his philosophy and understanding of life are absolutely different.
Second, one should in principle not discount somebody else's opinion based on the person, but judge it on its own merit. Just because someone was "troubled" doesn't mean that their view is worthless. (Especially when the cause of much of the trouble was a perception of the coming environmental crisis, leading him to be one of the first ecological activists.) In fact, as Grothendieck possessed both a unique and a supremely educated perspective, one should not be surprised if he has interesting things to say. At the very least, they will not be the same clichés that add no value.
Even in his writing outside of math, time and time again I have been amazed by how his perspective on a topic already accounted for my own. Experiencing this is certainly startling when you expect to be the one assessing the supposedly crazy views of another. His work is one of the very few that I have learned from upon reading multiple times, which is why I am very disappointed when people dismiss him based off of a stereotype.