This is such an interesting way to word this question just because it's usually stated in reference to Holden instead of Phoebe. Usually the reader wonders why Holden gets angry at Phoebe. Therefore it's interesting to consider your question in regards to the female character here. The general answer is that Phoebe doesn't want Holden to leave because he is her big brother and wants him to stay. However, let's expand upon that answer a bit in regards to the characters of both Holden and Phoebe.
Quite simply, Holden is Phoebe's protector. Both characters feel this way. Phoebe is Holden's little sister in every sense of the term. She is ten and full of life. Specifically, Holden wants to protect Phoebe from all the "phonies" of the world and from the evils of the world according to his perception of them. Holden really cares about Phoebe, possibly more than any other character. Note the joy in which Holden watches Phoebe on the carousel: a joy he experienced in his own youth.
Because of this "big brother" protection, it is a threat to Phoebe when Holden wants to leave and goes out west. Because Holden cares about Phoebe, that is why he shares his plans with her (not realizing this would wound her heart). Ah, a confirmation of Holden's thoughts:
Don't ever tell anybody anything. If you do, you start missing everybody.
Further, Phoebe decides that she wants to go with him. Holden reacts violently to this and some readers mistakenly think he begins "hating" Phoebe. This is not the case, it is simply a young man reacting to intense love of a sibling and trying to figure out how to handle it. Of course, Holden decides not to leave. This is when Holden finds joy in watching Phoebe ride the carousel at the zoo.
The long and the short of the whole thing is that Holden is a protector by nature and considers himself the true catcher in the rye. His relationship with Phoebe is no exception to this rule:
Anyway, I keep picturing all these little kids playing some game in this big field of rye and all. Thousands of little kids, and nobody's around - nobody big, I mean - except me. And I'm standing on the edge of some crazy cliff. What I have to do, 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 do all day. I'd just be the catcher in the rye and all. I know it's crazy, but that's the only thing I'd really like to be.