~에 익숙해지다 (~e ik-ssoo-kae-ji-da) – to become accustomed to something/to become used to something/to become familiar with something
This is actually an extension of 익숙하다, which simply means to be used to something.
The slight difference in nuance between the two is 익숙하다 is usually used to describe a state of being used to while 익숙해지다 is used to express how people become used to something.
In most cases, 익숙하다 is used as a descriptive verb (adjective) while ~에 익숙해지다 is used as an action verb.
Here are some simple examples:익숙한 일 (a work/task/job one is used to), 익숙한 사람 (a person one is used to).
익숙한 일 (a work/task/job one is used to), 익숙한 사람 (a person one is used to).
제가 이 일에 익숙해졌어요 (I got used to this work/task/job), 제가 이 사람에 익숙해졌어요 (I got used to this person).
제가 이 일에 익숙해지고 있어요 (I am getting used to this work/task/job), 제가 이 사람에 익숙해지고 있어요 (I am getting used to this person).
But you will realise that ~에 익숙해지다 is more commonly used in most situations because of the nature of the phrase and meaning of the words!
제 집에서 회사까지 너무 멀고 버스 타는 시간도 긴데 지금 출퇴근 길에 익숙해졌어요.
My house is really far from my office and the bus ride is really long but I got used to the commute.
어렸을 때는 내가 떡볶이를 좋아하지 않은데 지금 매운 음식에 다 익숙해졌어.
When I was young, I didn’t like spicy rice cakes but right now I became used to all spicy food.
한국어를 배우기를 시작할 때는 한국말 하는 것은 어색했지만 한국어 말하기에 익숙해졌어요.
When I started learning Korean, speaking Korean was awkward but I got used to speaking Korean.