할 수 있게 되는 is modifier form of 할 수 있게 되다 할 수 있게 되다 is combination of 할 수 있다 and 게 되다 할 수 있다 means "to be able to do" V+게 되다 means "to end up" By attaching ~게 되다 to the end of a verb, you can create the meaning of “to end up…”. Using this form indicates that the events that occurred happened out of one’s control, almost by chance or fate. For example: e.g. 저는 그 여자랑 사랑에 빠지게 되었어요 = I ended up falling in love with her 야구 경기에서 두산은 LG를 이기게 되었어요 = Doosan ended up beating LG in the baseball game 직장에서 더 멀리 이사하게 되었어요 = I ended up moving farther away from work 저는 그 여자를 일요일에 만나게 되었어요 = I ended up meeting her on Sunday 한국어를 할수 있게 되다 means "end up being able to speak Korean" 한국어를 할수 있게 되는 순간 means "the moment when you end up being able to speak Korean" 한국어를 할수 있게 되는 순간은 언제였어요 means When was the moment when you end up being able to speak Korean? =When did you end up being able to speak Korean? =When did you get to speak Korean? Refer to https://www.howtostudykorean.com/upper-intermediate-korean-grammar/unit-4-lessons-92-100/lesson-94/