The Stage as a tactical cycle serves mid-term coordination.
A Stage is a multiplication of sprints (E.g. six sprints are coordinated by a Stage). This gives the teams clearance for autonomously performance of four to six cycles within the team tactical cycle.
When a stage is introduced, this cycle is wrapped around the sprints. The sprints all have the same takt or rhythm. This allows for common events such as Common Sprint Review and daily scrum of scrums. In addition, the common takt reduces complexity.
In the Scaled Agile Framework (SAFe) the Stage is called a Program Increment.