Learn Hardcore Korean: naeilbuteo sugang sincheongi sijakdwae.

부터 marks the starting point in time or space. When attached to a time expression (like 내일), it means “from [that time].” You could contrast it with 까지 (“until”), as in 내일부터 다음주까지 (“from tomorrow until next week”).

What is 수강 신청, literally and functionally?

수강 신청 is a compound noun:

수강 comes from 듣다 (to take a course) + 강 (lecture), so it means “taking a course.”
신청 means “application” or “registration.”
Put together, 수강 신청 means “course registration.”

Why do we attach 이 to 신청, and why not 은/는?

이 is the subject marker for nouns ending in a consonant (신청 ends in ㅇ).
It simply marks “수강 신청” as the subject of 시작돼 (“registration starts”).
You could use 은/는 to add contrast or topic emphasis, but here 이 is the neutral subject marker.

What is 시작돼, and how is it related to 시작되다?

The base verb is 시작되다 (“to begin,” literally “to be started”).
In everyday speech, 되다
- -어 → 돼 (a common contraction of 되어).
So 시작돼 is the casual present form meaning “(it) starts” or “(it) will begin.”

How does the contraction from 시작되어 to 시작돼 happen?

시작되다 + 어 (present casual connective) → 시작되어
돼 is the spoken contraction of 되어 (dropping the ㅎ and merging vowels).
Without the polite 요, you get 시작돼 in informal speech.

What level of speech or politeness is this sentence? Why is there no 요 at the end?

시작돼 (without 요) is the plain informal style (반말).
It’s suitable among close friends, classmates, or in casual announcements.
To make it polite (존댓말), you’d say 시작돼요 or the more formal 시작됩니다.

How would you express the same idea more formally or politely?

Polite casual: 내일부터 수강 신청이 시작돼요.
Formal polite: 내일부터 수강 신청이 시작됩니다.