How much more praise deserved thy beauty's use, If thou couldst answer 'This fair child of mine
Here's where the sonnet changes gears.
We've been chugging along through some pretty depressing lines, and now all of a sudden, the speaker offers a solution to the problems of old age: "Have a baby!"
According to the speaker, the young man is wasting his beauty, not using it to produce anything. But, if he could put that beauty to work and crank out some kids, he would be deserving of real praise (line 9).
Shall sum my count and make my old excuse,'
Now we circle back to the imaginary question from line 5 ("Where did all of your beauty go?"). The speaker gives the young man a snappy comeback he can use when he's old. If people ask him what happened to his good looks, he can point to his kid and say: "Look, I passed them along to him."
Shakespeare uses a new metaphor here, and turns the kid into an accountant or a banker, someone who stores and keeps track of the "treasure" of youth and beauty that we heard about in line 6. If it seems like you've wasted your beauty, that you have been "thriftless" (line 8), showing someone your kid is a way to prove it isn't really lost. In that way, he "sums [the old man's] count" and clears his debts to the world.
Sorry to dork out here, but it's really cool to watch how Shakespeare will put an idea like treasure into the poem early on, and then wind his way back to it in subtle ways. Just one more reason why he's the greatest.
Proving his beauty by succession thine!
In this line, Shakespeare finishes up the theme of money and finance. Basically this line means that because this kid is your son, he has inherited your beauty.
That idea of "succession" (that just means "coming after") means that they own this treasure of beauty together, that the father's property has passed down to his son.
This were to be new made when thou art old, And see thy blood warm when thou feel'st it cold.
Here's the closer, the point that is meant to end the argument.
The speaker suggests that having a child is the closest thing to being born again, to being "new made" (line 13). Having kids becomes a way to cheat death. When you are old and worn out and your blood feels cold (line 14), you can look at your son and feel young and warm and beautiful and alive again.
Or at least that's the idea. We're not sure how well this argument worked on the young man, especially considering that Shakespeare wrote 16 more sonnets about this same thing.