Holden dreams of saving children from falling off a cliff.
...I have to catch everybody if they start to go over the cliff—I mean if they're running and they don't look where they're going. I have to come out from somewhere and catch them. That's all I'd do all day. I'd just be the catcher in the rye and all.
This is what he envisions when he thinks of the song mentioned in the title. Saving children from a certain doom, symbolically, relates to Holden's high regard for innocence and childhood.
Holden cannot face the loss of innocence, for reasons that are complicated, but not entirely obscure. He has lost a younger brother and continues to mourn for him, talking to his dead brother Allie often in times of stress.
Holden has lost something dear to him that he associates naturally with childhood and innocence. This can serve as a partial explanation for Holden's nearly pathological insistence on the preference of innocence and youth over maturity. (If maturity is defined by loss and death, Holden quite logically has chosen to resist this process.)
Holden's preference for innocence could not be clearer in the novel. He daydreams of going off to live in the pristine wilderness, untouched by development. He states that his younger sister is the only person he can talk to; his only friend and the only person he truly respects.
The antipathy Holden bears toward phonies and maturity is displayed throughout the novel and should be obvious.
Analyzing the reasons behind Holden's predilections, we can see that childhood, for Holden, represents a state of being before a fall (as demonstrated by his vision relating to the novel's title).
The park evokes his own fond memories of childhood, before his brother Allie's death, and seeing Phoebe circling around in this natural setting seems to bring him a sense of permanency and wholeness.