Think of it as a soliloquy -- the king, Ulysses, is looking back on his life while dealing with the commonplace of his old age. He says that his people don't really know who he is (or was).
It little profits that an idle king,
By this still hearth, among these barren crags,
Match'd with an aged wife, I mete and dole
Unequal laws unto a savage race,
That hoard, and sleep, and feed, and know not me.
Here he is describing his past life -- the travels, the wars, and all his adventures... how he lived life to the full. There are several references to his journey as chronicled in the Odyssey, which you should know -- you can look this up online here:
http://www.in2greece.com/english/historymyth/mythology/names/odysseus.htm.
I cannot rest from travel: I will drink
Life to the lees: All times I have enjoy'd
Greatly, have suffer'd greatly, both with those
That loved me, and alone, on shore, and when
Thro' scudding drifts the rainy Hyades
Vext the dim sea: I am become a name;
For always roaming with a hungry heart
Much have I seen and known; cities of men
And manners, climates, councils, governments,
Myself not least, but honour'd of them all;
And drunk delight of battle with my peers,
Far on the ringing plains of windy Troy.
Here, he is looking at his life philosophically. All memories fade and what is the use of his heroism -- what is life?
I am a part of all that I have met;
Yet all experience is an arch wherethro'
Gleams that untravell'd world whose margin fades
For ever and forever when I move.
How dull it is to pause, to make an end,
To rust unburnish'd, not to shine in use!
As tho' to breathe were life! Life piled on life
Were all too little, and of one to me
Little remains: but every hour is saved
From that eternal silence, something more,
A bringer of new things; and vile it were
For some three suns to store and hoard myself,
And this gray spirit yearning in desire
To follow knowledge like a sinking star,
Beyond the utmost bound of human thought.
He leaves his son, offers him some advice and then says he will have to live his own life.
This is my son, mine own Telemachus,
To whom I leave the sceptre and the isle,---
Well-loved of me, discerning to fulfil
This labour, by slow prudence to make mild
A rugged people, and thro' soft degrees
Subdue them to the useful and the good.
Most blameless is he, centred in the sphere
Of common duties, decent not to fail
In offices of tenderness, and pay
Meet adoration to my household gods,
When I am gone. He works his work, I mine.
He looks at what most of his life has been as a seafarer... calls out to his sailors and considers death as yet another journey.
There lies the port; the vessel puffs her sail:
There gloom the dark, broad seas. My mariners,
Souls that have toil'd, and wrought, and thought with me ---
That ever with a frolic welcome took
The thunder and the sunshine, and opposed
Free hearts, free foreheads --- you and I are old;
Old age hath yet his honour and his toil;
Death closes all: but something ere the end,
Some work of noble note, may yet be done,
Not unbecoming men that strove with Gods.
The lights begin to twinkle from the rocks:
The long day wanes: the slow moon climbs: the deep
Moans round with many voices. Come, my friends,
'Tis not too late to seek a newer world.
Push off, and sitting well in order smite
The sounding furrows; for my purpose holds
To sail beyond the sunset, and the baths
Of all the western stars, until I die.
It may be that the gulfs will wash us down:
It may be we shall touch the Happy Isles,
And see the great Achilles, whom we knew.
This is his final philosophy -- his advice to those who come after and his explanation of how he lived his life.
Tho' much is taken, much abides; and tho'
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are;
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.