Specific detail is often what makes writing immediate and real, yet Dickinson does not state what causes the “great pain.” Why does she leave out this detail?

While specific detail can make writing more vivid, it can also make it difficult for readers to relate when the detail described is completely outside of his or her experience. Much of Dickinson's writing draws upon the events of her own life, but here we might argue that the vagueness of the reference helps to make the poem more universal, allowing the reader to bring their own experience to bear in its interpretation. Dickinson does not refer to "my great pain," but instead describes a universal aftermath to pain, in general, and therefore pain as it is specific to each reader.
There are other details in the poem which are more specific and which help the reader to appreciate what the poem conveys. The aftermath of pain is described as "the Hour of lead," which, if survived, is remembered "as Freezing persons recollect the Snow." The language in the final stanza is suggestive of emotional numbness (leadenness), emphasized further by the imagery of "freezing persons" falling into "chill––then Stupor––then the letting go," which seems to allude to hypothermia. A sensory field is, therefore, created. Meanwhile, the generic articles applied throughout the poem to "the Nerves," "the stiff Heart," "the Feet" allow the reader to imagine their own nerves, hearts, and feet ("mechanical," again suggestive of numbness) in this situation.