"Prospice" means to look forward. In this poem, written in 1861, a few months after his beloved wife's death, Browning looks towards his own death, which seems to him nearer than ever before. He begins, first, by dwelling on his fear of death. He calls death the "arch" or highest fear.
He then remembers that he is a fighter. He decides that he must face and fight his fear of death bravely, like "the heroes of old." He must "bear the brunt" of this battle. If he does so, the "fiend voices" that cause him fear will "dwindle" and fade away. As they do so, he will be able to feel peace, then see light. He writes,
O thou soul of my soul! I shall clasp thee again,
And with God be the rest!
Browning thus affirms his faith in the immortality of the soul through the process of looking at death and facing his fear of it. He decides not to "creep past" death as if it won't come. When he faces it, he realizes his fears are unfounded: upon his death, his soul will meet up again with his wife's soul, already in heaven. His body might die, and he might suffer and struggle against death, but in the end, his soul will live, clasped to his beloved. Beyond that, he will leave the rest, and his final rest, in God's hands.