What does it mean when this card says that the assigned leaders stay at the system? For how long? Do i Swop one for another or potentially severa? Sounds wired. Or does it just mean that the assigned leaders stay on the board for a turn and dont go back to the pool?

From the normal rules for Rescuing Captured Leaders (from the Rules Reference):

After rescuing a leader with a mission, any leaders assigned to the mission may also move to the “Rebel Base” space. Any other leaders in the system stay in the system.

'For The Greater Good' overrides this, and does not allow the assigned leaders to leave the system at the end of the rescue mission. This is only for this mission; the refresh phase of the round will still return them to your pool.

The risk is that, between the mission and the refresh phase, the imperial player exploits this opportunity to capture this leader instead.