This pattern is used to express a final result that comes about from a change in state, opinion or plan that often goes contrary to the speaker's original expectations or is outside of the speaker's control. This commonly translates into English as 'it turned out that', 'things worked out in such a way that' or 'I came to understand/realize/believe/etc. that':
재미있게 됐어요. I turned out to be interesting.
벌써 나뭇잎이 누렇게 됐어요. The tree leaves have already turned yellow.
못 가게 됐어요. It's turned out that I can't go.
그 사람의 입장을 이해하게 됐어요. I came to understand that person's position.
내일부터 도서관에서 일하게 됐어요. I'm to begin working at the library tomorrow.
회사 일로 한국에 오게 됐어요. I have to come to Korea on company business.
Since '-게 되다' at times depicts that the way things turned out was outside the speaker's control, the pattern takes on certain politeness functions. Firstly, it can be used when the speaker wants to make clear that the action he/she took was unavoiable. As in the following examples, this is a particularly useful strategy when apoligizing:
자꾸 전화 드리게 되어서 죄송합니다. I would like to apologize for having to call you so often.
오늘 약속을 어기게 된 것 심히 죄송합니다. I am sorry for breaking off our engagement today.
In addition, the speaker may use this pattern to sound humble. By expressing the result as something outside of his/her control, he/she cannot be taken to be bragging or taking credit for it:
과장으로 승진하게 됐어요. It turns out that I've been promoted to department head.
국가 대표팀 선수로 뽑히게 됐어요. I have been chosen for the national sports team.
참고: Jaehoon Yeon and Lucien Brown(2011), Korean: A comprehensive Grammar, 251-252.