The capacity to “see beyond” basically means that a person is susceptible to memories. It is a special power, and it can mean different things for different people. Jonas sees colors, but The Giver heard music. In each case, the person is in touch with the memories of the past.
"When I was just a boy, younger than you, it began to come to me. But it wasn't seeing-beyond for me. It was different. For me, it was hearing-beyond."
Jonas frowned, trying to figure that out. "What did you hear?" he asked.
"Music," The Giver said, smiling. "I began to hear something truly remarkable, and it is called music. I'll give you some before I go." (chapter 20, p. 157)
In each case, the memory of the past somehow makes it through to the present with specific individuals, those who have the capacity to see beyond. It is important because these individuals, The Giver and the Receiver, are the only connection to the past. Without them and their abilities, there would be no memories at all.
The community relies on the memories to solve problems. They come to The Receiver for advice.
"They don't want to hear about pain. They just seek the advice. I simply advised them against increasing the population." "But you said that that was before my birth. They hardly ever come to you for advice. Only when they—what was it you said? When they have a problem they've never faced before. When did it happen last?" (chapter 14, p. 112)
The Receiver of Memory protects the community from pain. The committee members asked The Giver (at that time The Receiver of Memory) for advice and listened to his wisdom. They do not want to know where the wisdom comes from.
Jonas's ability to "see beyond" is important because it allows him to see color, which is an ability that other members of his community do not have. It is a character trait that is important to become the new receiver of memory.