In describing the girlchild as going “to and fro apologizing,” the speaker is implying that she has a marked lack of confidence in how she relates to other people. It's as if she feels she has to apologize for her very existence, as if the very fact of her existence is a standing affront to everyone she meets.
This is a common type of behavior among those with self-esteem issues. And in the case of the girlchild in the poem, one doesn't have to look very far to discover the source of her low opinion of herself. At the end of the opening stanza, we're told that a cruel classmate told her that she had a “great big nose and fat legs.”
Even though the girlchild has an awful lot going for her—she's highly intelligent as well as physically strong—her self-esteem begins to sink right after her classmate utters such hurtful remarks, so much so that she's constantly going around apologizing. What's happened is that she's effectively internalized her classmate's cruel words to the extent that she's come to believe them. All she sees, like everyone else, is a “fat nose on thick legs,” something for which she feels she should apologize.