As mentioned, 되게 means "very" and comes from 되다. But this 되다 is not the same one meaning "become", "make", or "turn into". There are homonyms 되다 and 되게 with an entirely different meaning. 되다다 means "thick" used for a battered mixture, paste, or dough used in cooking. If you don't mix or batter it right it become to "thick", with too little water in it. You say then "반죽(mixture) is 되다". You can easily see then why 되게 means "very". The adjective itself is sometimes used to describe a person too, like "tenacious" or "tough" in character. Also, this 되다 and 되게 are usually pronounced with lengthened vowel for emphasis while 되다 for "become" is short and light. Its antonym is 묽다 - "thin", "too watery".

Yes 되게 comes from 되다 which means to become, to be done or to make it. However, 되게 itself is just used to mean "very".