Reading poetry requires effort on the part of the reader. Before you can digest the words, you need to roll them around your mouth and chew them, as in the following quote, attributed to Stanley Victor Paskavich (emphasis mine):
Don't live by my words, don't die by them, chew them slowly digest them, and smile if they give nourishment to your soul.
Digestion as a metaphor for learning and assimilation is not new. See for example Francis Bacon's "On Studies" (1613):
Some Bookes are to bee taſted, others to be ſwallowed, and ſome few to bee chewed and digeſted.
If the "chewing" is taken literally, however, you end up eating (paper and) ink; as a result "The poems are gone" (line 7). This implies that poetry, standing in for "high culture", gets destroyed. At the opposite end of high culture are the dogs in the basement. They intrude into the library, which stands for rules (cataloguing rules, shelving rules) and order. From a psychoanalytic point of view, the basement can even be seen as a metaphor for the Freudian id, i.e. one's uncoordinated instinctual desires.
Poetry and the library "go to the dogs". The librarian is powerless against this backward development and hence "does not believe what she sees", "walks with her hands in her dress" and "begins to stamp her feet and weep". Man without culture becomes an animal, in this case a dog. He sees himself as a "new man", possibly because he considers himself freed from the shackles of culture. At the same time, however, the light first becomes "dim" (line 8) and is eventually replaced by darkness (18); on a literal level, this makes reading impossible, on a metaphorical level, the light of civilisation has given way to the darkness that represents the absence of culture.
Interestingly, the shaking off of the shackles of culture is not reflected in the poem's form: the poem consistently uses three lines per stanza until the end of the poem and the syntax remains regular (i.e. subject followed by predicate). Even though the "new man" "romp[s] with joy in the bookish dark", that darkness is still "bookish"; culture eventually still wins out.