The reason that the Giver says "forgive me" to Jonas after he transmits the memory is that the memory that he has given to Jonas is a very painful one.
In this memory, Jonas gets placed in the role of a very young soldier who is in the process of dying from wounds he sustains in a battle. Jonas experiences all of the physical pain that the soldier feels as well as his mental and emotional anguish.
Even after the soldier dies, Jonas continues to feel all of the horror of the battle that is going on around him.
The Giver knows that the memory that he is going to transfer to Jonas will be unique to him. Until now Jonas has only had good memories with the exception of sunburn. He has not experienced death and the horrors of war.
The Giver transfers the memory of the battle field to Jonas. Jonas sees the he young soldier dying and the fear that it includes. He is horrified for the first time. He knows the pain of death, fear, terror, and all the awful things that go with a war. It is a transition point for Jonas who now knows that death is a bad and painful experience.
Later when he sees his friend playing a war game, he tells him to stop.
Jonas does NOT like this memory at all because he continues to feel the horror around him.