This is a very important skill - for GCSE, above and below!
The first stage is to read the poem at least twice, first just going over it, and the second time underlining or highlighting things that you find interesting.
There isn't a specific method for how to analyse poetry, and I know it can be quite intimidating at times! It's important to have a checklist of things to look out for - these include form, rhyme scheme, imagery, and how the poem is narrated.
Some poetic forms are very famous, such as the sonnet, and it's useful to have a rough understanding of why and when these forms were used. For example, the sonnet, popularised by Shakespeare and his contemporaries, is often used for love poetry. However, a modern poet might use this ironically, to talk about the breaking down of a relationship instead. Similarly, the ballad is a popular form for medieval poetry, and so a modern poet using this form would be trying to evoke a sense of a time in the past, and maybe the qualities of chivalry and honour that medieval societies represented.
Additionally, sometimes how the poem looks is important! For example, some poems might look like dialogue from a play, or inconsistent line lengths could reflects the movement of the waves. It's very useful to use your imagination, within reason, to think about why the poet has made their poem look this way.
Some poems rhyme, and some don't! What's most interesting when thinking about rhyme is to think about exceptions. For example, if you have a non rhyming poem (sometimes called blank verse) and then a rhyming couplet at the end, the poet might be trying to emphasise the final two lines. Additionally, the words chosen for each rhyme could be important and connect to eachother.
Poems also use a variety of different tones - from dramatic monologues (when you have one figure talking / thinking as if in a play), the use of an objective narrator, or a dialogue between two or more people. Noticing which one of these has been used is also helpful, for example if you have a poem that is written in first person, it indicates that it might be emotional and personal. This would contrast a poem that seemed more objective and distant, for example the description of a place or event.
When poets are trying to describe something specifically, as if to create an image in the reader's mind, they will often use imagery. Sometimes, they will introduce imagery into their answer, and this could include the use of other images to describe something else. For example, using imagery of nature or animals to talk about a woman. Again, this might give the reader a sense of the tone of the poem.
The most important part of analysing poetry is to not be afraid if you don't understand what the poem is about! If that does happen, which is probably likely at some point, do still try and go through the specific techniques. Even if you can't make a clever comment about the meaning of the poem, an insightful remark about use of alliteration, form or rhyme is still very impressive!