By his own admission, Russell wasn't interested in aesthetics, so his comment is probably not part of a more elaborate philosophy. But...
"Anarchic and resistant to organization" perhaps conflates the two meanings of anarchy, one from British history, referring to a 17th century period of political chaos and civil war (an-arch meaning literally "without king"). This is the meaning predominantly used by mainstream politicians today.
However, anarchic in that sense has a purely negative connotation and presumably that is not what Russell intends, unless perhaps as a vague sentiment romanticizing chaos and disorder.1 The other meaning of anarchy refers to the political philosophy, of which Russell would certainly have been aware (he refers to Anarchists proper in the same document), and likely sympathetic to (as a leftist socialist) but which he probably did not have much first hand experience of. The political philosophy is critical of the role of arbitrary authority and leadership hierarchies in organization (again: "without king"), but it is not against organization (i.e., it is not a philosophy of disorder). Modern western representational democracies employ very strong authoritarian hierarchies, and it is often hard for their citizens to grasp a theory of political organization that resists such without considering it simply "chaos and disorder", hence conflating the two meanings is common.
Today someone like Russell would likely have been more sensitive to this distinction, but since it is really one that only started gathering steam in the early 20th century, he probably meant something involving the first meaning shot through the lens of the second -- i.e., a positive and not negative conflation -- and perhaps by "organization" he does mean some kind of authoritarian hierarchy.
Reading the text around the quote, however, it's not immediately clear who that organization is. It can't really be the Bolshevik government, since "while they controlled or destroyed the counter-revolutionary in all other social activities, they allowed the artist, whatever his political creed, complete freedom to continue his work". However, he seems to think this has actually created a problem, since apparently "the old artistic community [had] pronounced sympathy...with the old régime", leading to a schism: art which fairly explicitly either supports or opposes the state. Russell claims this situation "is death to the old individual art which depended on subtlety and oddity of temperament, and arose very largely from the complicated psychology of the idle", and while he is a proponent of Communism, he sees Communist art as essentially propaganda -- lacking in individualistic subtly -- and hence dead-on-arrival. This Romantic concern with the nature of the individual might be taken as an individualist anarchist stance.
In those pages Russell does in fact sketch a partial theory of aesthetics; note "arose very largely from the complicated psychology of the idle". Political activists are presumably not idlers and in post revolutionary Russia everything has been actively politicized: "The next generation in Russia will have to consist of practical hard-working men, the old-style artists will die off and successors will not readily arise."
This is really in regard to high art (the stuff that goes on in galleries and theatres), but there is another form of organization which Russell claims threatens what we might call low ("peasant") art, "the organization and development of industry, which is far more dangerous to art than Communist doctrine". The central point here is to do with mechanical reproduction:
The effect of industry on the crafts is quite obvious. A craftsman who
is accustomed to work with his hands, following the tradition
developed by his ancestors, is useless when brought face to face with
a machine. And the man who can handle the machine will only be
concerned with quantity and utility in the first instance. Only
gradually do the claims of beauty come to be recognized. Compare the
modern motor car with the first of its species, or even, since the
same law seems to operate in nature, the prehistoric animal with its
modern descendant. The same relation exists between them as between
man and the ape, or the horse and the hipparion. The movement of life
seems to be towards ever greater delicacy and complexity, and man
carries it forward in the articles that he makes and the society that
he develops. Industry is a new tool, difficult to handle, but it will
produce just as beautiful objects as did the mediæval builder and
craftsman, though not until it has been in being for a long time
and belongs to tradition.
All is not lost, it seems, but merely in turmoil. Note this seems to be dialectical reasoning.
The theory of aesthetics Russell presents here was probably not rigorously thought out (or intended to be taken as a formal theory), but the work is interesting as a commentary on a particular time and place, including the status of its art and artists.
1 As we shall see, he does have a very capital-R Romantic vision of the artist, and romanticizing disorder as a natural state unfettered by excessive civilization fits that too.