A Mind's identity is, in principle, information — and information can be copied. The pattern that constitutes a particular Mind, its mind-state, can be recorded and stored, so a Mind whose body is destroyed may later be restored from a backup, continuous enough with its former self to count, by the Culture's lights, as the same person.
The same technology underwrites the casual attitude to death that runs through the novels, reached by biological citizens through the Neural Lace. Continuity becomes a decision rather than a given. See also Subliming.