Holden hears what he wants to hear, not only with the lyrics of this song, but with everything.  He wants to "catch" children even when Phoebe doesn't want to be "caught."  Perhaps he doesn't want to think about the fact that HE is the one who needs to be caught or perhaps he doesn't realize until the end of the story that children cannot be saved from the cliff. 

In the novel The Catcher in the Rye, there are two references to the Robert Burns poem and the title of the novel.  In chapter 16, Holden sees a family walking together on the streets of New York City.  Holden sees a little boy, walking near the curb, singing "If a body catch a body coming through the rye".  It is at this time on the novel where Holden admits to his mental issue:

It made me feel better. It made me feel not so depressed any more.

The second appearance of the line is in chapter 22. Holden is speaking to Phoebe about getting kicked out of Pency.  He then mentions the "song":

You know that song "If a body catch a body comin' through the rye"? I'd like-
It's "If a body meet a body coming through the rye!" old Phoebe said.  "It's a poem by Robert Burns."

Here, Holden conveys his desire to be the one saving all of the children that try to run over the cliff.  The important thing here is that Holden's desires to save the innocent is at the cost of losing himself.