The capacity to see beyond is the ability to use any of the senses that were present in the memories, but not in the community. For Jonas, this ability reveals itself in his ability to see color.
Jonas does not think he is special. He thinks he is just an ordinary eleven year old boy. Even when he notices the apple change in mid-air, he does not understand why but does not think it is anything special about him.
But suddenly Jonas had noticed, following the path of the apple through the air with his eyes, that the piece of fruit had—well, this was the part that he couldn't adequately understand—the apple had changed. (ch 2, p. 24)
Later, Jonas is told by The Giver that he is beginning to see the color red.
Once, back in the time of the memories, everything had a shape and size, the way things still do, but they also had a quality called color. (ch 12, p. 94)
However, the Capacity to See Beyond seems to manifest itself differently in different people. Not all see the color red, or color at all, at first. The Giver tells Jonas that when he first noticed something different, he did not see color—he heard music.
"When I was just a boy, younger than you, it began to come to me. But it wasn't the seeing-beyond for me. It was different. For me, it was hearing-beyond." (ch 20, p. 157)
Jonas later sees all colors, and hears music. Basically, the Capacity to See Beyond means the ability to see into the community’s past, and receive their memories.
Lowry, Lois (1993-04-26). The Giver (Newbery Medal Book). Houghton Mifflin Harcourt. Kindle Edition.