Note: This analysis refers to the poem in its original language. Get ready for some Old English, people.
Okay, awesome readers, hold on tight, because we're about to get a little technical.
Unlike the poetry of say, Shakespeare, which has a predetermined number of syllables per line (ten), a line of Anglo-Saxon poetry doesn't have a set number of syllables. Instead, it has a set number of stresses (syllables with emphasis): four, with a slight pause in between the first two and last two stresses, called a caesura. The first stressed syllable of the second half-line has to alliterate with (have the same first letter as) one or both of the stressed syllables in the first half-line. We know we just threw a lot at you, so let's take a look at lines 48-49 of "The Seafarer" just so we can really see what we're talking about:
Bearwas blostmum nimað, byrig fægriað, (Groves take on blossoms, the cities grow fair,)
wongas wlitigað, woruld onetteð; (the fields are comely, the world seems new;)
These lines have different number of syllables (11 and 10), but in each case there are four, stresses, a pause between the first and second half-line, and alliteration among the stressed syllables, with the b of berwas, blostmum, and byrig in line 48, and the w of wongas, wlitigað, and woruld in line 49.
It's worth noting that in the translation we've used, the translator makes quite an effort to use alliteration in modern English. Of course it doesn't quite have the same effect, because the stressed syllables are not the same. But it's nevertheless a nod to the repetitive sounds that would have been obvious to Anglo-Saxon ears.
If you're interested in developing some Anglo-Saxon ears of your own, you might want to learn more about alliterative verse, which is the technical term for the form we've just described. Other famous Anglo-Saxon poems, like Beowulf and "Caedmon's Hymn" were written in it, too. Plus, you'll be the center of attention at your next dinner party.