Grieving is one of the characteristic activities of human beings. Although it seems that some other species (e.g., members of the corvid family) experience sorrow, grief is a more complex phenomenon. Recently I had occasion to read a fine book on the topic by Michael Cholbi, entitled Grief: A Philosophical Guide. Here are the essentials of what I gleaned.
To begin, we must understand that grief is not an emotion but an activity. Yes, grieving can involve a wide range of emotional reactions: sorrow of course, but also anxiety, guilt, anger, even joy (as Epicurus said: "Sweet is the memory of a dead friend."). But grieving also includes mental states such as confusion and perplexity, as well as mental activities such as deliberation and reflection.
The grounding for this intellectual component is the fact that any significant personal relationship entails a conception of what someone means to you - and that kind of meaning is bound up with your identity, your social role (e.g., within a family), indeed your entire conception of what it means to be human.
Cholbi makes a twofold argument. First, when someone close to you dies (or leaves you or otherwise disappears from your life), the relationship - formed of memories, expectations, longings, commitments, characteristic activities, and so on - is still there, but vastly changed. Second, because of the aforementioned ties to personal identity and practical meaning, coming to understand those changes is a process of gaining greater self-knowledge and thus represents a philosophical achievement.
Perhaps Emily Dickinson summed it up best:
There is a finished feeling
Experienced at Graves -
A leisure of the Future -
A Wilderness of Size.
By Death's bold Exhibition
Preciser what we are
And the Eternal function
Enabled to infer.
Finally, it's worth noting that grief was actively discouraged by most of the ancient philosophers, including Socrates, the Stoics, Chuang Tzu, and the Buddha; two notable exceptions were Aristotle and Confucius.
(Cross-posted at philosopher.coach.)
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