leaving the nest; fledging
Literal use for a young bird leaving its nest. Often appears with birds such as ツバメ or ひな.
Our family watched the swallows leave the nest.
going out into the world; becoming independent
Figurative use for a child, student, or young person leaving a protected environment and beginning an independent life.
Graduation day is the day the students go out into the world.
As a parent, a child's becoming independent is both happy and lonely.
At the graduation ceremony, the children reach the moment of leaving the nest.
Today is the day my son leaves the nest.
The teacher watched over her students as they went out into the world.
Spring is the season when swallows leave their nests.
自立 focuses on the state of being able to stand on one's own and support oneself, while 巣立ち focuses on the moment or event of departing into that independent life.
卒業 literally means graduating from school and is used narrowly for completing an educational stage, whereas 巣立ち evokes the broader image of leaving home and can describe any transition into independent life.
A noun form related to 巣立つ, from 巣 meaning 'nest' and 立つ/立ち in the sense of setting out or rising from a place. The spelling directly reflects the literal image of leaving the nest.