At the most basic level, the user clicks on the picture to indicate where the "active" control point should be moved, and an image is returned reflecting the shape of the curve with the control point in the indicated location. (The rules the computer uses to draw these curves is described in "How Snowflake Works".)
The "Redraw" button is useful at any time when the user wants to change a setting and redraw the picture without changing the location of the control points. For example, one might wish to change the number of iterations with the "Iterations" popup menu, or change the shape or visibility of the grid, etc.
Note:The grid is "there", whether is is visible or not--if you have
"Snap To Grid" checked, it will work even if you have "Plain Background"
selected.