expression, ichidan verb
Literal botanical sense: a plant puts out buds or sprouts.
暖かくなって、庭の木に芽が出てきた。
As it got warmer, buds started to appear on the tree in the garden.
種をまいて一週
The seeds sprouted a week after sowing.
expression, ichidan verb
get lucky; have luck on one's side
Figurative idiom: luck finally comes one's way, often after a period of waiting or effort. Related to 目が出る (sense 1), but this word uses the kanji 芽.
See also: 目が出る
長い下積み生活の末、ようやく彼にも芽が出た。
After years of obscurity, luck finally smiled on him.
地道に続けていれば、いつか芽が出るかもしれない。
If you keep at it steadily, you might get lucky someday.