A symbol or metaphor is a concrete thing which represents a concept more abstract. It can be intrinsic or explicit (implied or explained). Also, its degree of interpretation (relation between symbol and concept) can be restricted to a particular culture or it can be universally understood. It often has multiple meanings. For example, a flower could represent fragility, love or life; the sea could be seen as the source of things (return to the womb); its horizon, broadmindedness or the call to adventure. Such is the case in this story.
In Johnathan Livingston Seagull, the metaphors used are implicit; they are suggested rather than explained. They are also universal by nature. (This is one reason this book has been so popular.)
At first Johnathan is part of the crowd; he is indistinguishable from the other birds on the beach but at least he belongs. His yearning for flight, though, makes him leave the group and finally fly. Johnathan represents the individual and the flock of seagulls, collectivity or society. The beach - conformity and its limitations. His distancing from the others represents his need for individuality and the other birds' mocking of him, his subsequent alienation.
When Johnathan finally learns how to fly, he just doesn't take off - he soars. The euphoria and power he feels at this point compensate fully for all his effort, loneliness, even the rejection of others. His flight represents freedom; the limitless sea, his new horizon.