What's he saying?
"O! how thy worth with manners may I sing, / When thou art all the better part of me?"
How can I tactfully praise you when, since our lives are conjoined, I am praising part of myself?
"What can mine own praise to mine own self bring? / And what is't but mine own when I praise thee?"
What good is it to praise myself, since I'm not gaining anything that wasn't mine already?
"Even for this, let us divided live, / And our dear love lose name of single one,"
So let's live separate lives, and no longer think of ourselves as one person,
"That by this separation I may give / That due to thee which thou deserv'st alone."
So, thus separated from you, I can fully praise you without praising myself, too.
"O absence! what a torment wouldst thou prove, / Were it not thy sour leisure gave sweet leave,"
Separation would be horrible, since you would have free time,
"To entertain the time with thoughts of love, / Which time and thoughts so sweetly doth deceive,"
Permitting me to kill time thinking of you, tricking myself,
"And that thou teachest how to make one twain, / By praising him here who doth hence remain."
Absence allows for the creation of a partner through the poet's praising in this sonnet of the fair lord, who is far off.
Why is he saying it?
Sonnet 39 is one of the "separation sonnets," and is tied closely to Sonnet 36, which begins, "Let me confess that we two must be twain, / Although our undivided loves are one." Sonnet 36 can be read as if it were spoken by the young man, or his "advocate," who is the poet himself, on the young man's behalf. The fair lord insists upon separation, "Lest my bewailed guilt should do thee shame;" he does not want to affect the poet negatively anymore by having his offense extend to the poet. In this interpretation, Sonnet 39 is a belated response to the idea of separation.
In the first 8 lines, the poet agrees with the idea of separation wholeheartedly. His reasoning is that while the two men are together, their love makes them as one, and thus in praising the fair lord the poet seems to heap praise upon himself. That is useless and impolite (impossible to do "with manners"), so the separation is a good idea since it will leave him free to praise the fair lord as much as he deserves. The word "alone" can mean without the speaker, or it can be interpreted as praise that only the fair lord deserves among people.
But in the second half of the sonnet, the poet is struck by the gravity of separation, and grapples with the idea of filling his time alone. Line 9 begins with an address to "absence," personifying the idea; this addressee is referred to again in line 13, "And that thou teachest how to make one twain," with "thou" being the idea of absence, which makes one person into a pair of people. But the word "twain" also means "separated," so the distance is alluded to even in this hopeful line.
The purpose of writing about the fair lord has changed since earlier sonnets. There, it was to record the beauty of the man in his youthful state, so that even if he did not bear children, he would be immortalized in verse. Now, it is not for the sake of the world, but for the poet himself that he writes of the young man. Since they are separated, it helps him imagine that he is still in the presence of the fair lord to write about his beloved.
The idea put forth in line 2, that the fair lord is "the better part" of the speaker, is commonly used by Shakespeare to refer to a soul mate. For example, in Comedy of Errors, Antipholus of Syracuse says to Luciana, "It is thyself, mine own self's better part, / Mine eye's clear eye, my dear heart's dearer heart," in Act III. But it could also mean the soul as opposed to the body, as it does in Sonnet 74: "The earth can have but earth, which is his due; / My spirit is thine, the better part of me." Either reading reveals how deeply the poet feels connected to the fair lord, and thus how painful is the separation.