What is the meaning of the poem "String Quartet" by Carl Dennis?

The poem discusses the poet's desire for fulfilling conversation. To him, good conversation is like a finely tuned quartet or orchestra. No one tries to talk over anybody else, and everyone is allowed to add to the conversation. There is no disharmony and no conflict when friends converse respectfully with each other.
In fact, the poet contends that a conversation isn't likely to flourish "When sales technicians come trolling for customers,/ Office-holders for votes, preachers for converts." Here, the poet suggests that religion, politics, and the competitive marketplace are often areas fraught with conflict and strife. He says that, even though there are likely "good people among such talkers," their conversation can never match the well-tuned and disciplined musical discourse of a harmonious quartet. In a good quartet, voices/musical instruments are never at odds; they are never pulled away from their true purpose of giving pleasure to listeners.
The poet talks about his admiration for the quartet in the third stanza. He cites how the cello is often "occupied/ With repeating a single phrase while the others/ Strike out on their own." Yet, invariably, all the four instruments eventually come together again at a later juncture. The reintegration is seamless, and this is what the poet admires about a musical "conversation."

Each near the end swerves back to the path Their friend has been plodding, and he receives them As if he never once suspected their loyalty.

In the fourth stanza, the poet wonders how he will react if he discovers that harmonious conversation can be had in both the musical world as well as the world of human discourse. He says that the quartet may be telling him something: that "conversation like this is available/ At moments sufficiently free and self-forgetful." Basically, people can have the same harmonious and satisfying conversation in both worlds. Also, we are more likely to have fulfilling conversation when we concentrate less on what we want to say ("free and self-forgetful") and more on how we can add to the experience.
In the last stanza, the poet comforts himself that, no matter what happens, he can at least revel in the role of the listener. Listeners are "glad for what they manage to bring to the music/ And for what they manage to take away." Basically, we can learn much from just listening while others talk. Additionally, we add to a conversation when we are good listeners.