안다=hug but 버리다 does not mean 'discard/cast away' in this phrase. In Korean (해)버리다 used after some other verb, say 안다, means ' sb has done it without any intention, by accident. so, 안아버렸어 means ' I hug her(him) in spite of myself'

In the above situation, 버리다 is used an assistant verb. so the 버리다 means that the action of the front word안다 already ended. And its result you may feel sorry or in adverse you become relived. 동생이 케익을 먹어 *버리고* 없다 when your sister ate the cake you spared right before you got home. Feel sorry? 말을 해 *버리니* 시원하다 right after you confessed your feeling to the boy you like. Feel sorry or relived?