"Sonnet 116" begins with a vow: the speaker of the poem promises—to himself and to the reader—that he will not "admit impediments" to the "marriage of true minds." The fact that the speaker begins the poem with a vow raises some questions. A reader could imagine the poem almost as a prayer: something the speaker says to himself, in privacy, as he ponders the beauty and majesty of true love. Alternatively, a reader could imagine the poem as a passionate declaration: something the speaker says to the person he loves.
In the latter case, the poem and its meditation on love become more complex as readers think about the speaker's motivations. One might wonder why, under what circumstances, a statement as grandiose, beautiful, and moving as "Sonnet 116" would be necessary. In other words, what is the speaker responding to?
Both readings the poem—that the speaker is simply thinking to himself, or that he is addressing a specific romantic partner—are possible. A major challenge for interpreting this poem, then, will be deciding why the speaker says the things he does—and to whom.
In either case, as he begins his vow, the speaker uses a series of rather ambiguous phrases. The reader may wonder what actually constitutes a "marriage of true minds"—or what it would mean "to admit impediments" into such a marriage.
The speaker uses the word "marriage" in the normal way: the legal union between two people. But also he uses it in a metaphorical sense: to refer to a dedicated union between people or things, independent of wedding vows. The use of the word "minds" at the end of line 1, rather than, say, "people," underscores this metaphorical sense: "minds" can't get married in a church, but they can be closely joined together in love or friendship.
This metaphorical sense is potentially key to understanding the poem. The first 127 of Shakespeare's sonnets are addressed to an aristocratic young man with whom the poet has an intimate relationship. The young man and the poet can't be legally married, but their relationship might be profound enough to justify being described metaphorically as a marriage.
At the end of line 1, the speaker lays out the conditions for such a marriage: the parties involved must be "true minds." In other words, they must be true to each other.
In line 2, the speaker concludes his vow, hoping that no "impediments" will interrupt the relationship between these "true minds." The word impediments strongly recalls the language of the marriage service in Renaissance England, as outlined in the Book of Common Prayer, in which the minister would say to the two people getting married, "I require and charge you ... that if either of you do know any impediment why ye may not be lawfully joined together in matrimony, that ye confess it." The opening lines of the poem reenact this marriage ceremony—staging, if only in the speaker's fantasy, a marriage between him and his beloved. One can imagine the opening lines of the poem as the speaker's response to the minister: he does not (and will not) admit any impediments.
It's worth noting that, at the heart of these wedding vows, there is a requirement to confess: the bride and groom must tell the minister if there's a problem. The speaker imposes a similar demand upon himself, asking himself to "admit" any impediments. Here, the word "admit" means two things at once: first, to concede or acknowledge the existence of something; second, to allow something to enter. This means that the speaker's vow might be interpreted in two ways.
First, he will not acknowledge any impediments in the marriage of true minds—because no such impediments exist. Second, he will not allow any impediments to poison or transform this marriage going forward. Most likely, both interpretations are valid and both senses are at work at once. The speaker is simultaneously enacting a marriage between "true minds" and expressing his highest hopes for that marriage—that it will continue unchanged, indefinitely. In either case, the enjambment between lines 1 and 2 reinforce his point: his sentence flows past the impediment of the line break, without pausing to observe it.
The first line of the poem is metrically ambiguous, though the poem overall is written in iambic pentameter. The first line of the poem could arguably be read as follows:
Let me not to the marriage of true minds
This clearly doesn't align with perfect iambic pentameter, perhaps sonically reflecting a kind of "impediment" that the speaker argues love will overcome.