To leave the nest — young birds or animals leaving their parents' nest; can be used literally for animals.
春になり、ツバメのヒナが巣立った。
In spring, the swallow chicks left the nest.
巣立ったばかりの小鳥が初めて飛ぶ姿を見た。
I saw a young bird that had just left the nest fly for the first time.
ヒナが元気に巣立てるように餌を与えた。
I fed the chicks so they could leave the nest in good condition.
To go out into the world / become independent — metaphorically leaving one's family/home to start life in society (e.g., graduating or starting work).
息子が大学卒業を機に巣立って一人暮らしを始めた。
My son left the nest and began living alone after graduating from university.
若手社員が次々と巣立って、会社に新しい風が吹いた。
Young employees left one after another, bringing fresh energy to the company.
学生が社会に巣立つ準備をするための研修が行われた。
Training was held to prepare students for leaving school and entering society.