In The Giver, the Chief Elder tells the crowd that the Receiver has told the committee of Elders that the new receiver must have the capacity to see beyond. She openly states that she does not know what it is. Jonas has been seeing slight glimpses of color in this black and white world and he tells the crowd (once he has been selected) that he thinks what he sees is beyond.
As it turns out, he is partially correct. To put in modern terms, Jonas has the ability to see "outside of the box". He can see beyond what is his reality and imagine another world- a world that is totally different from his own. Perhaps more importantly than his vision, seeing beyond refers to his thinking and his ability to grasp concepts beyond that which he has been taught and trained.