The speaker's refrain shows off her bravado and self-confidence—and, the poem hints, helps her to feel a little braver.
Over and over across the poem, the speaker repeats the same line: "Life doesn't frighten me at all." Those words become the poem's heart (as well as its title): this speaker, the insistent repetition shows, is one brave kid.
Sometimes, the speaker even uses a slightly different version of her refrain to emphasize just how brave she is. For instance, when she mentions the dangers of "panthers" and "strangers," she changes her phrasing up a little, saying, "No, they don't frighten me at all." Here, she seems to be saying: Nope, no matter what you name, I'm not even a little scared of it!
Of course, maybe repeating this refrain isn't just a way for the speaker to tell readers how brave she is, but also a way for her to feel braver. After all, she lists plenty of scary things before she insists that "They don't frighten me at all." Her refrain thus starts to sound a lot like a spell or a battle cry, a phrase that gives her the strength to confront everything that does frighten her.
In other words: the refrain perhaps helps the speaker to banish every fear she names.