And dig deep trenches in thy beauty's field, (line 2)
The worst thing about age is that is destroys beauty. In this line the speaker is making a big deal about wrinkles, trying to get his reader worked up. He exaggerates the seriousness of aging, turning lines on someone's face into deep trenches. This is definitely the kind of strategy that would work best on someone who is vain. If you aren't obsessed with your own looks, aging might seem natural. If you are vain, though, aging might seem awful, and it would be easy to get you freaked out. Once that's done, you are more likely to do anything you can to avoid the consequences of getting old – which is exactly what the speaker wants.
Thy youth's proud livery, so gazed on now, (line 3)
The idea here is that the young man's expensive clothing (and everything else about him) is so beautiful that everyone loves to look at him. However, the word "gaze" is interesting. We detect (and we're not the first to do so) just a little bit of longing in this line. It's as if the speaker of the poem has some personal experience with what it's like to gaze on this young man. The note of attraction isn't too loud in this particular sonnet, but the speaker certainly spends a lot of time dwelling on this guy's looks.
How much more praise deserved thy beauty's use, (line 9)
One important idea in this poem is that beauty isn't just something that you have that eventually goes away. Beauty is something that you can put to use, that you can invest, and that you can pass along. If you "use" your beauty well, it will be a treasure that you can pass along to others. If don't use it, it will just waste away.
Proving his beauty by succession thine! (line 12)
This is the goal. Beauty doesn't have to die, but can be passed along like a shiny car to your son. You get the pleasure of looking at it, and you can also say, "Hey, look! That used to be mine, and now it belongs to my son!" Now remember, this poem is being addressed to someone who seems to care a lot about appearances, so it makes sense to boil down all the pleasures of being a parent to one easy-to-swallow package.