The poem's title readers know what the poem will be about: renouncement refers to the act of giving up something one enjoys or desires. The poem itself begins with the speaker announcing exactly what it is she intends to let go of:
I must not think of thee; [...]
"Thee" sounds formal, but it just means "you." The speaker, for reasons that aren't revealed in the poem itself, has chosen to give up fantasizing about someone she clearly longs to be with.
Alice Meynell wrote the poem after falling in love with the Catholic priest who helped convert her, and the poem reflects the fact that this love was never going to be realized (priests take vows of celibacy and can't marry). The speaker's motivations in the poem are left vague, however, allowing the reader to imagine any number of possible scenarios. Perhaps this "thee" is already married; perhaps they just don't love the speaker back. What's clear is that thinking about this person won't do the speaker any good.
The firm caesura created by the semi-colon in the middle of line 1 makes the speaker's opening statement sound utterly resolute. She might be "tired," worn out by the effort it takes not to think about this person, but she's determined to get them out of her head:
I must not think of thee; and, tired yet strong,
She escalates things in line 2 when she declares, "I shun the thought that lurks in all delight." The verb shun means to forcefully, vehemently avoid or reject something. The anaphora of lines 1 and 2 ("I must," "I shun") make the speaker sound more emphatic still. She actively, consciously rejects this "thought"—the "thought of thee," despite the fact that it "lurks in all delight."
In other words, this thought sneakily hides within everything that brings the speaker joy: "in the blue heaven's height," or up in the bright blue sky, and "in the sweetest passage of a song." Basically, everything lovely or happy thing reminds the speaker of her beloved. The breathy alliteration of "heaven's height" and the gentle sibilance of "sweetest passage of song" make these lines themselves sound lovely. It seems that the speaker enjoys thinking about this person, which makes her efforts to stop thinking about them all the more difficult.
A Petrarchan sonnet, "Renouncement" consists of 14 lines of iambic pentameter. This is a meter in which each line contains five iambs, poetic units that follow an unstressed-stressed syllable pattern (da-DUM). Here are lines 1-2 as an example:
I must | not think | of thee; | and, tired | yet strong,
I shun | the thought | that lurks | in all | delight—
This firm, propulsive meter reflects the speaker's confidence and resolve. And yet, already, there's a subtle irony in these lines that undermines that resolve. The speaker uses apostrophe here and throughout the poem, directly addressing the person she is trying to push from her mind. On the one hand, this apostrophe makes the poem sound more intimate and vulnerable; the speaker is quite love-sick, though trying very hard to take control of her feelings. At the same time, this apostrophe undercuts the speaker's claim that she's not going to this person; she's talking to them directly, and has written a whole poem about them!