The concept of -(으)니' is when something in the first clause reaches to the point of the second clause. Depending on the context, meanings vary.
(1) The first clause provides temporal context for the second. For example:
12시가 되니, 배가 고픕니다.
Now that it's 12 o'clock, I feel hungry
되다 -> 되 + 니.
해가 떴으니, 일어나야지요.
The sun came up; it's time to get up.
뜨다 -> 뜨 + 었(past tense) +으니
(2) The first clause becomes a reason for the second clause. Colloquially, modern Koreans also use '-(으)니까' more to mean this. I use '-(으)니' quite a lot, and my younger native Korean friends tell me I sound aged. :\
Anyway, examples:
시간이 없으니, 빨리 준비하세요.
We don't have time, so hurry up and get ready.
없다 -> 없 + 으니
오늘 늦게 일어났으니, 늦게까지 잠이 안오겠다.
I slept in this morning, so I won't get sleepy until late tonight.
일어나다 -> 일어나 + 았 (past) 으니
I hope this helps!