I think you have to approach this question not just in terms of the words related to how to tell the time but in the viewpoint of how koreans read the time.
we say "it's a quarter to 4." in English.
it means it's a quauter minutes towards 4 o'clock.
we say "4시 십오분 전입니다." in korean.
it means it's a quarter minutes before(or backward from) 4 o'clock.
it's like we are standing on the long hand(at 45 minutes' point) looking toward 4 o'clock in English.
In Korean, a quarter minutes is seen backwards from the viewpoint of standing on the top of the hour. so we use the word, '전', which means 'before'.