One mechanical reason to break levels up into multiple parts is to provide a mental break for the player, allowing them to reset their concentration or to provide a clear pausing point for them in case they need to put the game down and attend to other aspects of life.
A similar reason would be to provide more of a sense of accomplishment and progression. Providing goals to complete that are part of a larger goal, even if the scope of both is small, provides both positive reinforcement to the p layer upon completing each smaller goal and a bit of a carrot to keep the player engaged when it looks like the next goal is close at hand. It helps with that "well I'll just finish one more task before I put this down" feeling.
Another reason to break levels up into stages is for thematic reasons, to have a clear transition between environments, ideas, or concepts (but one that still fits within the overall theme of the outer level structure). This is more common in narrative-based games; you might also see cutscenes used to break up these stages.