You can earn Star Flowers by replaying Stages within a Chapter where all the White Flowers have blossomed and by earning more than the required number of points for those Stages.
Furthermore, Star Flowers affect your Ranking status and grade.

How to bloom Star Flower
1. Clear all the Stages in a Chapter with a White Flower
2. Clear all the Stages in the Chapter again, with a higher score to bloom Star Flowers
(The scores to bloom Star Flowers differ in every Stage.)

Rankings
1. 1 Star Flower increases your Rankings status by 1
2. 1 Star Flower increases your Flower grade number by 1