The Receiver places his hands on Jonas's back and transmits the memory of sledding down a hill: Jonas does not just remember the activity, he feels the cold air and the snowflakes against his face and the thrilling speed of sledding downhill.
Physical contact is necessary for transmittal. Exactly how memories are "transmitted" is never explained, making them seem magical and extra-powerful.
Afterward, The Receiver tells Jonas that other people do not have the memories of sledding or hills or snow because the community gave up these things on purpose. Hills made it hard to move goods and snow made it hard to grow food, so the communities became climate-controlled. He says that these memories came from the time before "we went to Sameness."
In order to make life easier, more comfortable, and more stable, the founders of the community had to sacrifice pleasurable experiences like sledding. To make these losses bearable, the people gave up any memory, for them and their descendants, of what they lost.
Jonas wishes aloud that hills and snow and sledding still existed, and asks why the Receiver, with all his power, doesn't bring them back. The Receiver responds that he has honor, which is a very different thing from power.
The people of the community aren't interested in the Receiver's memories or wisdom. The Receiver has given up trying to assert his power.
The Receiver next gives Jonas a memory of sunshine. At Jonas's request, he then gives Jonas his first memory of pain—a mild sunburn. Jonas is startled by the sensation, but begins to understand that painful experiences are necessary to be able to appreciate the pleasant ones.
The community chose to give up pain and hardship for stability. But through his first painful memory, Jonas starts to understand that one must feel pain in order to feel joy.
As Jonas leaves for the day, The Receiver tells Jonas not to call him "The Receiver" any longer, since Jonas is actually the new Receiver. Instead, Jonas should call him The Giver.
The change in names is a passing of the torch. Jonas is assuming his new role.