At the beginning of the story, Jerry strongly desires the acceptance of these boys. The narrator tells us
They were of that coast, all of them burned smooth dark brown, and speaking a language he did not understand. To be with them, of them, was a craving that filled his whole body [....;] as he preserved his nervous, uncomprehending smile, they understood that he was a foreigner strayed from his own beach, and they proceeded to forget him. But he was happy. He was with them.
These boys are older than Jerry, and he very much wants to be on their level. Their cool turn-taking and nonchalance about swimming underwater through the rock frightens Jerry and makes him feel different from, lesser than, them. The tunnel through the rock can be read as a symbol of the passage to adulthood, and these "big boys -- men to Jerry" pass through it with ease, whereas Jerry knows there is no way he could even attempt such a feat now. He understands, after his fruitless attempts to draw their attention back to him, that
They were leaving to get away from him. He cried openly, fists in his eyes. There was no one to see him, and he cried himself out.
Like a child, he cries when he sees that he is not equal to them, not accepted by them as a peer.
By the end of the story, however, Jerry has made a plan in order to be able to swim through the rock as the "big boys" have done. He has practiced and worked hard, and still he holds off attempting the feat because he wants to be sure he is really prepared. The narrator says, "A curious, most unchildlike persistence, a controlled impatience, made him wait." Jerry is beginning to develop from a child into an adult. He now looks back on the beach he used to play on with his mother as "a place for small children [...]. It was not his beach."
In the end, after he has literally bled and nearly died to achieve what the older boys accomplished so easily, "He did not want them." Jerry has realized that he's not quite ready to be one of the big boys yet; he's not ready to be an adult (though he has begun the process of developing into one). This realization is further supported by Jerry's childish behavior with his mother in the final lines. Calling her "Mummy" (as a child would), and clearly hoping for her praise, he blurts out that he can now hold his breath for almost three minutes. He no longer seeks the approbation of the older boys; he is content with his mother's praise, as a child would likely be, because he recognizes that he just isn't ready to be "a man" yet.