First, loosely screw the strap back in. Don't screw it in too tight, or you'll have trouble getting the spring in. Once it's screwed in, adjust the mechanism so that it's lined up with one of the pegs.

Place the spring so that the coil is around the nearby spoke, and the lower arm is seated in its little notch behind the strap screws.

Put pressure on the coil of the spring so that it doesn't fly out, and move the arm of the spring over the top of the fastener, and push it in so that it stays underneath the lip.

While keeping pressure on the coil, quickly tighten the screws. Tighten the screw closest to the spring first. This will prevent the spring from unseating itself and springing back out.