For Holden Caulfield, the best thing about the Museum of Natural History is "that museum was that everything always stayed right where it was."
He adds the following:
"You could go there a hundred thousand times, and that Eskimo would still be just finished catching those two fish, the birds would still be on their way south, the deers would still be drinking out of that water hole ... Nobody'd be different. The only thing that would be different would be you. Not that you'd be so much older or anything. It wouldn't be that exactly. You'd just be different, that's all."
This contributes to one of the novel's overall motifs: Holden's fear of change. Throughout the novel, Holden attempts to protect the children around him, particularly Phoebe. In fact, the title of the novel The Catcher in the Rye is an ironic mistake by Holden. Holden mistakes the words in the poem, which say "If a body MEET a body coming through the rye" and replaces this with "catches." Then Holden goes on to explain that he wants to be the one who "catches" kids before they fall off a cliff. This is ironic because the poem is about a boy and girl meeting out in a field of rye to probably have sex, something Holden seems deathly afraid of.
The reason this relates to the museum is that everything in the museum is preserved. Holden does not have to do any "catching" here.