In The Giver who does Jonas give the memories to and why did he give the memories?

Jonas gives memories to Gabriel to keep him calm.

The ability to receive and give memories is unique in Jonas’s community.  A person has to have the Capacity to See Beyond in order to be receptive.  We know that this is a very rare gift because the community avoids having most citizens in contact with memories.  Jonas has the ability, and he soon realizes that Gabriel has it too.

Families in the community are only supposed to have two children—a boy and a girl.  Jonas’s father is able to get a special extension for Gabriel and brings him home instead of releasing him.  Perhaps Jonas’s father is attached to Gabriel because he reminds him of Jonas.  Both boys have the same rare trait of light colored eyes.

Jonas becomes attached to Gabriel too.  One day Jonas begins to rub his back when he is fussing, and accidentally transmits a memory to him.  Up until this point, doing this had never occurred to him.  He was the Receiver, not the Giver, of memories.

He was not aware of giving the memory; but suddenly he realized that it was becoming dimmer, that it was sliding through his hand into the being of the newchild. Gabriel became quiet. (Ch. 14)

Jonas is shocked and surprised when he realizes what he has done.  He understands that Gabriel is special like him.  The light eyes are a clue that they are related, and it seems that this means that Gabriel could be a Receiver of Memory.

Jonas uses this ability to help Gabriel when he takes him and runs.  When Jonas realizes that Gabriel is threatened, he takes off with him.  He has come to see Gabe as a little brother and loves him like a brother.  He understands that Gabe is sensitive and special, things the community does not understand.  On the run, he uses memories to soothe and sustain the baby.

Before he had left the dwelling, he had laid his hands firmly on Gabe's back and transmitted to him the most soothing memory he could: a slow-swinging hammock under palm trees on an island someplace, at evening, with a rhythmic sound of languid water lapping hypnotically against a beach nearby. (Ch. 21)

Throughout the dangerous journey, Jonas uses Gabe’s ability to receive memories to help keep them alive.  He gives him memories of food and warmth when they do not have the real thing.  In the end, they both survive—but barely—due to the help the memories have given them.

Jonas and Gabriel share a special connection, and it is one that most people in the community would never understand.  They do not know emotions or love.  They realize that it is a sad thing to release a Newchild, but to them it is a disappointment and not a travesty.  Other than The Giver, Jonas alone appreciates the real emotions behind caring about someone.