As with any poem, literature speaks uniquely to each reader.
...was composed of open wounds. Vulnerable, tormented, and troubling...
This is the case in "Lobster." Note that Sexton personifies the lobster as "he." Perhaps she identifies with its sad story: a victim it would seem.
The first line, "a shoe with legs," describes how the poet sees this sea creature as it walks. It is not a beautiful or elegant shoe, it would seem, but perhaps a man's shoe—lonely: abandoned or lost in the water. Consider "A stone dropped from heaven"... rather ordinary. It generally does not draw much attention, the same way a stone is overlooked: it's grey and dull. What elevates this lobster beyond a simple stone is that is has been "dropped from heaven." "Dropped" brings to mind a mistake. While it can often reflect specific intent, it seems (to me) that it may not really belong here. It was not "flung" or "thrown" as stones often are, but it was "dropped." And having arrived, it travels along "mournful" and "alone." This makes me believe that its separation from heaven weighs heavily on it. For if we are to believe what we have heard or read about heaven, coming to Earth cannot be the best of the two worlds.
"He is the old prospector for golf" is not as easy to understand. Sexton does not say he is "a" prospector, but "the" prospector. For some people, golf seems a mindless game of chasing a small white ball endlessly over green lawns, to hit it into a small cup with a skinny stick that varies by degrees in its uses. To others, golf is a game of extraordinary concentration and finesse. The ball sometimes seems to have a will of its own. The sport commands silence and it creates suspense. How, then, does the lobster fit in with golf? Perhaps he is searching for a perfect place to rest himself. Perhaps he is wishing for an existence that is more fulfilling. "Prospector" denotes a continuous effort; it may also infer an endless task—a goal never realized.
A "God-head" is defined as "the essential being of God." In that it is plural, Sexton may be alluding to the different names by which the "Supreme Being" is referred all over the world. Dreaming of "God-heads and fish heads" may tie in to the earlier reference of heaven, as he dreams perhaps of returning to the place from which he was dropped. However, as a lobster, his physical condition would require food: fish heads.
The "cradle" that "fastens around him" may seem innocent enough to a creature that lacks intelligence. A "cradle" is used to protect a child at its most vulnerable time: while it is sleeping. However, "fastens" is an emotionally charged word: there will be no leaving this kind of "cradle." Sexton clarifies this imprisonment: it is "trapped." We know that it is waiting in U.S. waters, though we can assume that while personified, it is unaware.
It is nighttime: the world turns, with cigarettes, cars and robberies—all alien to the lobster. His only intent has been to move on.
Now the movement of the poem dramatically shifts. He is an "old hunting dog"—perhaps it is as if he is being put down because he can no longer take part in the hunt. For in the morning, he will be lifted and "undrowned," but soon boiled so his "perfect green body" turns red: the color of his death.
This poem may speak to Sexton's own sense of feeling out of her element, not understanding the world around her, and controlled and destroyed by the very things she cannot comprehend.