To analyse something means to look closely at its smaller parts and determine how they work together, and to link its function with the history of things like it. For poetry, that means looking (and listening) closely to its words. “Poetry is concentrated word magic” by which is meant that poetry is a concentrated, efficient utterance using all the properties of words (denotation, connotation, pronounciation, history, syllabification, rhymes, etc.) to evoke something in the reader/listener that is more than simply a logical, deductive, non-magic communication.
Practically, poetry analysis can begin with a categorization process: In what category does the poem belong? This is determined by several factors: subject matter (paean?, elegy?, epithamion?, etc.), line arrangement (sonnet?, couplet?, quatrain? etc.), rhythm of line (iambic pentameter?, trochaic quatrameter?, etc.), rhyme scheme (abab, aabb, aababbcb , etc.). Then, analyse the effect of these mechanical devices on the “running” of the poem’s mechanism—its flow, its movement from one idea to the next. Then analyse the way each word choice works toward the poem’s function—why “weak and weary” instead of “powerless and fatigued”? Why “Something there is that doesn’t love a wall” instead of “There seems to be something in Nature that doesn’t tolerate walls”? The analysis goes on, looking (and listening) at every detail, until the “workings” of the poem are revealed.
As can be seen, the analysis can be very complex (like dissecting a frog to see how its digestion, reproduction, etc. work), but should result in a renewed appreciation of the “magic” that results. Of course, sometimes (as when a child takes apart a clock to see why it “clicks”) the poem’s luster and life will be lost (as a biologist would say, “We murder to dissect”).