Like many poems by Heaney, this poem is rich with imagery, and uses that imagery to make evocative observations about the world in which people live.
As far as how we can analyze it, start by reading it a couple of times. First read it simply and literally: what does it say? Then go over it again, observing patterns and techniques.
The first extended stanza looks backward on a shared memory that sounds innocent. The verb "would ripen" indicates that this experience of picking blackberries was not something that happened once, but something that repeated. The sheer expanse of the descriptions in this stanza paint an almost Edenic (Eden-like) state of pleasant innocence.
There are hints or traces of potential darkness earlier, when the berry juice is like "summer's blood," but the mood really shifts in the final four lines of the first stanza. That's where the imagery turns darker. The berries become like eyes, and the pickers' hands are bleeding and bloody. The pierced hands echo stigmata, but the reference to Blackbeard makes them less victim than criminal.
The darkness accelerates in the final stanza, moving from images of death to images of rot. Here is where the themes really become explicit. This is not just a pleasant memory about a good time. This is a memory of a lesson the poet learned time and again. Time passes, especially good times. You can't save the sweetness of these berries, or of life, against time's power. Time always moves on. Things always rot. This is the lesson of aging, and of humans being powerless against entropy. As the poet says, "It wasn't fair." And it never is. But it is reality.