Learn Hardcore Korean: hyudaepone bimilbeonhoreul seoljeonghaeyo.

In this sentence, 에 marks the target or location “on” which you configure something—in this case, your phone. You use 에 when you want to say “set/configure X on Y.”

에서 would emphasize where the action takes place (“I’m at the phone setting”), but here you’re focusing on the phone as the object being configured.

What does 비밀번호 literally mean, and how is it constructed?

비밀번호 is a Sino-Korean compound:

비(非) = not
밀(密) = secret
번호(番号) = number
So literally “secret-number,” i.e. “password” or “PIN.”

Why do we use 설정해요 here? What does 설정하다 mean?

설정하다 means “to set,” “to configure,” or “to establish.” In tech or daily life, you use it whenever you choose or customize a setting:

컴퓨터에 폰트를 설정해요. (I set a font on the computer.)
알람 시간을 설정해요. (I set an alarm time.)

Why is -를 used after 비밀번호?

-를 (or -을) is the object marker in Korean. It tells you that 비밀번호 is the direct object of 설정해요—the thing being set.

Could we omit 에 and just say 휴대폰 비밀번호를 설정해요?

Yes, in casual speech you sometimes hear that omission. It still makes sense because “휴대폰 비밀번호” clearly shows what you’re setting. But formally or in writing, including 에 is more precise.

How polite or formal is the -요 ending in 설정해요, and how can we change the register?

-해요 is the standard polite style (존댓말).
To be more formal (e.g. in announcements), use 설정합니다 (합쇼체).
To be casual with friends, you might say 설정해 (반말).
To make a polite request: 설정해 주세요 (“Please set…”).

Can I say 핸드폰 instead of 휴대폰? Are they interchangeable?

Yes, both mean “mobile phone.”

휴대폰 (Sino-Korean) feels a bit more formal.
핸드폰 (loanword from English “hand phone”) is very common in everyday conversation.

Could we use 걸어요 instead of 설정해요 when talking about passwords?

Koreans often say 비밀번호를 걸다 (literally “hang a password”), but this usage is more colloquial. In formal or technical contexts, 설정하다 is preferred.