The connotative meaning of the word "heart" is passion. Unlike denotative meaning, which is the literal meaning of a word as defined in a dictionary, connotative meaning is the meaning that depends on the emotional context. So, a heart has one connotation when we speak about a surgery, and a totally different connotation when we talk about a broken heart, for example.
In this poem, the speaker tells us how "they" coerce a man to give all of his being to the political cause that he might or might not believe in. In the aforementioned line, the man has to give his breast, heart, and shoulders - his soul, passion, and strength, therefore everything he has, as if it were a commodity.
I am 100% sure about this.
