The in-game descriptions are more extensive than they first appear. Select Options (X), then View Help. Here there are additional screens of text (indicated by a triangle at the bottom-right) which may be accessed by pressing (A) or by tapping the screen.
To answer the original question:
To capture a star, it must be covered with the same color of Tetrimino from both sides.
The color of the star indicates what color covers it on the other side; early stages start out with it covered from the back so you need only cover it from the front. If a star is not covered on either side, it is white.
If you cover a star with the wrong color, the “front” and “back” matrices are exchanged, so that you start playing what was the “back” matrix. You then need to clear the occupied space of the old color and place a piece to match the color you just placed on the other side causing the flip.