Please follow these transformations that keep the meaning preserved, and I think you will get the reason: 1a. Spero di esserti stata utile 1b. Spero che io sono stat? utile a te 1c. Spero che io sono stata utile a te /io/ is the speaker: a woman

it's right for a woman: "stata" is feminine. Past participle is unvariable if the auxiliar verb is to have (but wioth the direct object pronouns). If it's to be, the participle is like an adjective: io sono stato al mare lei è stata al mare noi siamo stati al mare Lucia e Maria sono state al mare

You use 'spero di esserti sta utile' when someone has done something for you, for example if someone has helped you and then he wants to know if his help has been useful for you. So know that i've answerde you to expalin it to you I can say :<Spero di esserti stata utile>