Matthew Arnold is using an extended metaphor in comparing human beings to islands in "To Marguerite—Continued." A metaphor is a comparison that does not use the words like or as. An extended metaphor is used throughout a poem and usually expresses a theme.
Arnold compares humans to islands because islands are isolated pieces of land, surrounded on all sides by water. They are separated wholly from other pieces of land. Arnold's speaker uses the metaphor of humans as islands to express the theme of loneliness as well as to help us visualize his belief that humans are wholly alienated from other humans beings.
Arnold's speaker contrasts the reality of human isolation—being an island—with his belief that humans were once united, saying that
Parts of a single continent!
His exclamation point emphasizes his longing for that oneness to be the case again, because he doesn't like the feeling of being alone.
The speaker ends up blaming God for human separation, stating that "God their severance ruled!" This is an allusion
to the fall of humankind in the Garden of Eden, when Adam and Eve ate the forbidden fruit that divided them from unity with God and each other. The island metaphor almost certainly alludes back to John Donne's meditation in which he declares that "no man is an island." In fact, Arnold's speaker says that we are all islands (he uses the archaic term "enisled":
Yes! in the sea of life enisled,
With echoing straits between us thrown,
Dotting the shoreless watery wild,
We mortal millions live alone.