"A Birthday" celebrates love's arrival. What specific type of love is open to interpretation; it sounds like the speaker is celebrating romantic love, but the poem might just as well be praising the birth of a child or the love of God. Whatever specific form this love takes, it has completely transformed the speaker's world.
In these opening lines, the speaker compares their heart to various delightful things in order to illustrate how happy they are. First, they declare, "My heart is like a singing bird / Whose nest is in a water'd shoot." In other words, the speaker is as delighted and content as a bird twittering away in the safety of its nest.
The nest sounds like it's in a good spot too, resting "in a water'd shoot." A shoot refers to a young branch, something freshly grown, thanks to that nearby "water[]." In addition to simply conveying the speaker's joy, then, this simile evokes the speaker's rebirth: the speaker is "water'd" by love. The image of a nest might also symbolize home and domestic intimacy—things the speaker perhaps feels they've finally found in their beloved.
Lines 3 and 4 then follow the same structure as lines 1-2, creating anaphora and parallelism as the speaker uses another simile to convey their feelings:
My heart is like an apple-tree
Whose boughs are bent with thickset fruit;
The repetitive nature of the language creates a sense of abundance, as though the speaker's heart is so happy that they could think of any number of similarly happy things (or, perhaps, that no simile is quite right; nothing can really capture the bliss the speaker feels!).
This time, the speaker compares their heart to an apple tree with branches so heavy with fruit that they're "bent." It's simply bursting with fruit, just as the speaker's heart is bursting with joy. The image of "thickset fruit" suggests plenitude and might even call to mind reproduction (i.e., being fruitful). The bold alliteration of "boughs"/"bent" evokes the sheer weight of those branches, loaded up with apples ripe for the picking.
"A Birthday" is written in iambic tetrameter, meaning each line contains four iambs: poetic feet that follow an unstressed-stressed pattern. Here are lines 1-2 as an example:
My heart | is like | a sing-| ing bird
Whose nest | is in | a wa- | ter'd shoot;
The gentle pulse of those iambs creates a confident, soothing rhythm.