How does a check spring on sewing machine work?

The purpose of the check spring is to maintain tension on the thread as the thread take up arm returns to the top of the stroke pulling the thread back out of the cloth.

The take-up arm follows the needle. As the needle leaves the cloth, the friction between the thread and the cloth forms a loop that the hook inside the machine catches, pulls around the lower/bobbin thread forming the locked stitch. As the take up arm returns to the top of the stroke the thread between the tensioner and the take up arm becomes slack. The check spring pulls on this slack to keep the thread taut.

Proper stitches can not be formed if the check spring is not set with sufficient "snap-back" or tension on the spring. Setting the "snap-back" is by feel. Adjust it from loose to tighter to get good stitches. It is a three way balance between the upper thread tensioner, the check spring and the bobbin tension.