Execution Context Saving means saving the point in time context of all data and process states to the Process Server Database, if a server crash happens or a restart is needed the BPD/Task will be restored to the last saved execution context. It happens at the following three
1. When transitioning from BPD to Service Engine
2. At each Coach
3. At any point where Save Execution Context is ticked
The Coach save execution context checkbox is greyed out for a coach meaning it can not be disabled
Unless you clearly understand the need and use of save execution context it should be disabled for all other steps, as a performance tuning method you can run a process server database query to find out where all it is checked and then disable them. The more heavy your business object load is more resource intensive the save execution context will be, so it is a good idea to null out your large objects e.g. database result set objects etc.