To admit authorities, however heavily furred and gowned, into our libraries and let them tell us how to read, what to read, what value to place upon what we read, is to destroy the spirit of freedom which is the breath of those [library] sanctuaries.

The excerpt you ask about is taken from the latter portion of the Virginia Woolf essay, "How Should One Read a Book?" With a faulty logical premise as its basis, that being that literary judgement (i.e., literary criticism) and literary taste are contradictory and mutually exclusive approaches to reading [see top quotation], this essay makes the case for reading literary works--whether prose or poetry, fiction or fact--from the mind and perspective of the author, as far as is possible.

Do not dictate to your author; try to become him. Be his fellow-worker and accomplice.

In this essay, Woolf enumerates ways in which personal taste, preference and experience affect reading and take precedence over the rules of critical judgement, rules determined by "authorities." This might mean moral authorities (e.g., clergy) or literary authorities (e.g., critics; "Coleridge and Dryden and Johnson"). Woolf stresses the opposite of rules for reading. She stresses that the "sanctuary" of the library must be free of external influence; must not be bound by rules of how, what, and why one reads:

[Rules] destroy the spirit of freedom which is the breath of those sanctuaries ....

In the quote you ask about, Woolf is elaborating upon her thesis that rule-free reading begins with identifying with--sympathizing with--the author. In this quote, she acknowledges the counter argument that--even if you do identify and sympathize with the author and have an open mind ("if you open your mind as widely as possible ...")--there remains a part of the human brain that judges and evaluates and chooses to understand based upon personal experience rather than upon identification with the author. In other words, the counter argument says, where we should embrace the world presented by an author, we judge that world and say yea or nay to it, loving what we read or hating what we read. Woolf ascribes this to human nature: "we cannot sympathise wholly or immerse ourselves wholly ... we learn through feeling; we cannot suppress our own idiosyncrasy without impoverishing it":

We may stress the value of sympathy; we may try to sink our identity as we read.  But we know that we cannot sympathise wholly or immerse ourselves wholly; there is always a demon in us who whispers, "I hate, I love", and we cannot silence him.

Woolf offers a possible solution to the asserted dichotomy between taste and rules by suggesting we "train" our taste (i.e., preferences) and develop some control over our taste, rather than be moved by sheer energy of sentimentality. Thus our taste might eventually lead us to see deeper qualities and extended relationships between books (which, ironically, is a definitive part of literary criticism, thus seemingly a flaw in her argument):

as time goes on perhaps we can train our taste; perhaps we can make it submit to some control. ... Thus, with our taste to guide us, we shall venture beyond the particular book in search of qualities that group books together; ....