"오후에 구름이 끼기 시작했어요."
Translation:It started getting cloudy in the afternoon.
-기 is a particle that gets added to verbs to convert them into nouns. You can think of it similar to -ing in English or -ando/endo in Spanish
A lot of words/grammar principles require nouns in front on them, so 기 is used a lot. You'll see it a lot in V기 때문(에) = because of V기, or in V기 시작하다 = To start doing V기
Also if you use any computer/Internet stuff in Korean language, then you'll see a lot of buttons with Verb+기 written on them, which tell you what the button does. It's very common.