In the first stanza of "When We Two Parted," the speaker describes his reaction to the end of a relationship with his beloved. The fact that he is "silent" indicates that he is beyond despair, that he can't even protest or try to win her back. If this poem is autobiographical, referencing Byron's secret affair with Frances Webster, this makes sense because she has been with another man. And if their affair was secret, he must remain quiet (silent) about it. Within the context of the poem, it is unclear if the affair is between the woman and the speaker or if it is between the woman and another man. In either case, the speaker reacts with great sadness and he is silent because he feels defeated; there is no way to save the relationship. What makes it even more difficult for him is that she acts with cold indifference. His sadness grows because she doesn't seem as deeply affected as he is.
The "half broken-hearted" line could mean that, because she does not seem broken-hearted, only half of the heart they shared is broken (his half). It might also mean that his heart is severed in two. Or it could mean that his heart is initially half-broken and will become completely broken over the course of the succeeding years.