In "Blackberry-Picking," Heaney tackles an age-old poetic idea: we're all going to die eventually. But he does it in a pretty relatable way. By showing us the rotting berries (and the desire to keep them from rotting), he reminds us of how we cling to life in spite of our knowledge of death. It's pretty moving. We all want to keep what won't stay; the fact that these blackberries are impermanent or temporary is what makes them so precious. So the berries, because they're perishable, are something to treasure in the speaker's eyes.
The desire to collect all of the fresh berries is an attempt to take in the most of life and, in turn, is an effort to keep death away.