When you're a small child in the process of growing up, one of the things you do is integrate together all your emotional states and brain functions into a single cohesive identity for yourself. If something very bad happens to you at that age, sometimes the process of integration doesn't happen properly, and you end up with two or more emotional states that don't feel like being the same person, with different sets of memories and different sets of cognitive abilities accessible in each. That's DID. There's a few related dissociative disorders (such as OSDD-1B) which are fairly similar, but differ in what's shared and what's separate between the different emotional states.

Also, some people seem to end up in a state similar to DID from prolonged overuse of certain drugs (e.g., ketamine) or by having a childhood that wasn't quite bad enough to develop DID followed by something very bad happening to them in adulthood.