1) Arrange the jaws with their small ends facing up (spring seats towards the table).
2) Install the springs into the seats.
3) Grab the body of the chuck and place it over the jaws. Use your fingers to gently tip the front edge of all three jaws into the chuck body.
4) Push the chuck body towards the table. This should capture all three jaws without popping the springs out of their seats.
Sometimes, it can be helpful to install a small drill, say 1/4" into the chuck body as you push down towards the table. This prevents the jaws from tipping too far and spitting out the springs. Might have to experiment a couple times to get it right.
Yeah, two of those springs are a little trashed. If the chuck runs true, it's fine. But if they get bent enough the jaws no longer stay 120 degrees apart and the chuck starts to run out.