Faileas,
On an unboosted, non-irreversable elevator control aircraft, the elevator will desire to fair with the local airflow.
If the aircraft is out of trim in pitch (for speed change or cg change reasons), we will have to hold some elevator displacement to compensate.
If you rotate the big surface that the elevator is attached to (the horizontal stabilizer) we change the net downforce generated by the HS. If we rotate the HS (trim) in the proper direction, we can take out the need for elevator displacement.
By rotating the HS, we maintain the net downforce supplied by the HS & elevator combination (and trim out any corrective elevator deflection we may have had to apply before trimming.) The angle between the HS and elevator change, and the yoke or stick will move slightly in the process.
The HS is now carrying all the downforce and the elevator is neutral (faired with the local airflow).