When forty winters shall besiege thy brow, (line 1)
At the beginning of this poem, the passing of time is a pretty scary thing. It attacks you, freezes you, carves you up and wears you away. Time isn't a series of happy, carefree summers, it's a long stretch of winters that slowly pulls you down. More than anything though, it steals away your youth and beauty, and leaves you with nothing. The poem has to set up this evil, scary version of time so that the speaker can convince the young man to do something about it.
Thy youth's proud livery, so gazed on now, Will be a tatter'd weed, of small worth held: (lines 3-4)
This is another example of how time is a killer. It takes the beautiful, fresh clothes you wear when you are young and turns them into crummy old rags. Maybe Shakespeare is partly talking about actual clothes (a beautiful suit getting old and tattered over time) but the general idea is that your whole body and your life gradually become worn out, and life moves from pride and happiness to shame and sadness.
Then being ask'd where all thy beauty lies, (line 5)
Now the poet asks the reader to put himself into the future and imagine the kind of conversation he will have in forty years. Apparently people will wonder why he isn't good looking anymore, and he won't have much to say to them. In a sense then, there is some time travel going on in this poem. The speaker asks us to imagine the way we'll look and feel when we are old, and even what kind of conversations we'll have.