Help もらいます as auxiliary

Hello!

もらいます is used with the て-form in the following sentence:

I have learned about the use of もらう as an auxiliary with the て-form to express receiving a favour or benefit from someone, or to express politely getting people to do something.

In this sentence, however, I don't see how "I will change my appointment" fits in with the meaning of the もらう usage. Am I missing something?

Thanks!

That's because it's not "I'll change my appointment", it's "I'll have (someone) change my appointment for me."

Presumably, plans were made, and changing the date/time is going to result in an inconvenience on the part of the other party, so the speaker is receiving the benefit of having plans changed (note that it's 約束, not 予約, although there can be some overlap in the meaning) for their (the speaker's) benefit.

The subject, i.e., the recipient of the favor is not the agent (the doer of the action) in the expression ～てもらう. The agent is indicated by に.
e.g.
彼女は彼に本を買ってきてもらいました。
subject/recipient of the favor: 彼女
agent/the one who bought a book for her: 彼

In your example, the subject is the speaker, so they are not the agent/the one who changed the plan, as jt_-san explained.

Incidentally, ～させてもらう is used when the speaker is the agent. This expression works as a humble expression, so it's often used in business scenes.
e.g.
お客様のために快適なお部屋をご用意させていただきました。（＝ 用意しました）

Excellent explanations, both. Thank you very much.