Choice A is the best answer. The narrator explains that Mrs. Quabarl told Lady Carlotta about the "nature of the charge" when she gave Lady Carlotta details about the Quabarl children (line 53-61"During...century"). Since Lady Carlotta is pre-tending to be a governess, the term "charge" refers to her responsibilities, or job duties, when caring for the Quabarl children.
Choices B, C, and D are incorrect because in this context "charge" does not mean attack, fee, or expense.