How I have suffered grim sorrow at heart, have known in the ship many worries [abodes of care], the terrible tossing of the waves where the anxious night watch often took me at the ship's prow, when it tossed near the cliffs. […] (4-8)
The ocean on which the speaker travels is a dangerous place. Those tossing waves don't just throw the ship out of control – they do it "near the cliffs," where there's a danger of running aground and springing a boat-sinking leak. Right away, at the beginning of the poem, the natural world is a dicey, frightening place.
[…] Fettered by cold were my feet, bound by frost in cold clasps. […] (8-10)
The cold and frost totally mean business. They've got our speaker shackled, which is an interesting contrast with his description of himself as a traveler in constant motion. He may be physically in motion, but he feels trapped by his environment.
[…] I, wretched and sorrowful, on the ice-cold sea dwelt for a winter in the paths of exile, bereft of friendly kinsmen, hung about with icicles; hail flew in showers. […] (12-17)
This passage links together a lot of important themes of the poem: exile, sadness, loneliness, and, of course, extreme winter weather. Being in exile is no cakewalk. In fact, it seems like the most miserable place ever: bad weather, no friends, and icicles.
[…] There I heard nothing but the roaring sea, the ice-cold wave. At times the swan's song I took to myself as pleasure, the gannet's noise and the voice of the curlew instead of the laughter of men, the singing gull instead of the drinking of mead. […] (17-22)
The speaker's focus on these birds shows just how desperate he is for companionship. Even though those birdsongs are a poor substitute for the laughter of men, he's willing to listen because that's pretty much all there is to hear.
[…] Storms there beat the stony cliffs, Where the tern spoke, icy-feathered; always the eagle cried at it, dewy-feathered; (22-24)
These bird cries aren't the wistful sounds they were in lines 17-22. Now they indicate the struggle for life of both tern and eagle as the eagle seeks to make a meal of the smaller bird.
The shadows of night darkened, it snowed from the north, frost bound the ground, hail fell on the earth, coldest of grains. […] (31-33)
It's getting downright nasty out there. This description of worsening winter weather occurs just before the speaker launches into a description about how he feels troubled. So maybe, just maybe, the arrival of the storm signals the arrival of another storm inside our speaker. The weather acts as a barometer of what he's feeling.
Groves take on blossoms, the cities grow fair, the fields are comely, the world seems new: all these things urge on the eager of spirit, the mind to travel, in one who so thinks to travel far on the paths of the sea. (48-52)
If you thought all of the speaker's problems were related to the terrible weather, this passage just might prove you wrong. Even nice weather inspires our guy to get out on the whale road. Here, the connection between the natural world and our speaker's inner state seems a bit more distant.
So the cuckoo warns with a sad voice; the guardian of summer sings, bodes a sorrow grievous in the soul.[…] (53-55)
These birds just will not shut up. Because its song marks the arrival of warmer weather, the cuckoo becomes a symbol of the time for travel. The speaker does not necessarily want to travel, so its voice sounds "sad" to him. This passage is a perfect example of the way the speaker's mindset influences his interpretation of the natural world around him.