In the poem "The Seafarer," the use of alliteration gives the piece a lyrical and musical sound when read aloud, as was the custom for songs, dramas and poems—before the concept of literature existed, before things were written down. As these types of pieces were created and adapted over time, sometimes they changed in content or presentation as they were shared in the oral tradition (spoken or sung aloud) by scops (storytellers) around campfires, in castle halls or in taverns. It is worth noting that, since the poem has been translated from Old English, the specific examples of alliteration we find in modern translations might not necessarily have been present in the original Old English.
Literary devices of all kinds lend themselves to what the listener's ear catches—long before books or recordings existed. The imagination was the storyteller's best friend. Engaging the imaginations of the listeners was a skill that a scop developed over time; often he or she would also use a musical instrument as accompaniment—such as a lyre or small harp.
Alliteration is the repetition of (generally) a consonant sound taking place at the beginning of a group of words.
[It] is the repetition of an initial sound in two or more words of a phrase, line, or sentence.
If you are familiar with "Peter Piper picked a peck of pickled peppers," you are familiar with alliteration—with the repetition of the "p" sound found at the beginning of the words, clustered together.
One example from "The Seafarer" is found in the following:
...Wither and they mourn the memory of friends.
Notice the repetition of the "m" sound in mourn and memory.
In the following three lines, the anonymous author uses the "s" sound repetitively:
The sons of princes, sown in the dust.
The soul stripped of its flesh knows nothing
Of sweetness or sour…
Another example is:
No givers of gold...
The "g" sound is repeated.
To be able to appreciate how the spoken word can come alive, read "The Raven" by Edgar Allen Poe, and then listen to a recorded version of it.
Poems were originally meant to be read aloud, often as poets competed with each other to see who could write better poetic verse. Use of sound was an essential part of the experience.
Adventures in English Literature, Harcourt Brace Jovanovich Publishers: Orlando, 1985.