We are first clued into the identity of the speaker from the title. But we can infer her gender from pronoun usage and syntax within the first several lines. By line 5 we know the cause of her sorrow: exile. These introductory lines drip with lament, preparing for the emotional ride down the dark, cavernous rabbit hole of sorrow that will ensue for the rest of the poem.
Though the emotions are clear, however, the plot is anything but. Scholars today still struggle with the Old English original, and no two translations are alike. When reading the poem today, we lose much of the historical context that might have helped us make better sense of the story. But while the ambiguity in the narrative continues to yield a wide array of interpretations and theories, the tone is unmistakable. Without a doubt, the most striking aspect of "The Wife's Lament" is its elegiac voice; the speaker's heartache, expressed through lovely, plaintive language, is what we remember.
Tone and narrative aside, there's one more question on everyone's mind: who wrote the poem? There's tremendous speculation over this question. Some parties believe it was a man, possibly a monk, writing in a woman's voice, while others believe it was actually a woman. The fact of the matter is that there's no concrete evidence either way. You'll have to decide for yourself.