I need to rewind the spring and starter pull cord. I need to understand how all the parts fit together

I had the same problem. I had to take it apart and rewind it manually. To do this:
Step 1:
There are 4-bolts to remove on the orange plastic section the screw into the motor housing (one of them is hidden inside the orange plastic housing, you'll see an opening for it). There are also 3 screws in the black plastic exhaust housing that screw into the orange plastic housing, find those and remove.

Step 2:
Carefully remove the engine from the plastic housing. I didn't disconnect the throttle cable or the electric wires (points) just kind of worked around them.

Step 3:
The mechanism is kind of crappy. There are two black pieces with a spring in the middle, and a metal disc on the bottom with a ribbon coil on the underside of it. The top plastic piece is what turns the flywheel when you pull the cord. The spring in between connects the two plastic piece. This is probably what popped out. There is a piece sticking up on one end, which inserts into the hole in the top black plastic piece. If you remove the little guardrail things you can probably remove just that piece. KEEP TENSIONS ON THE PULL CORD. Once you remove that top black piece, it should start to recoil.

Step 4:
If it did recoil, put the spring back in the top plastic piece, and then put the plastic/spring combo back into the the bottom plastic piece, there is a little groove for the bottom of the spring to slide into. Then put the little guardrail screws back on. Make sure the big metal wire thing hasn't slipped down into the shaft of the weed whacker. That needs to go back into the motor, as it is what gets spun and in turn spins the strings. As you start to put the motor back in, you may have to pull on the pull cord a few times to get the flywheel and plastic piece to line up. Once you hear the motor start to catch, then you should be able to wiggle it back into place and put back all of the screws. Then give it a shot. If it pulls out again, you may have to drill a new hole into the top black plastic piece for that spring to slide into (this is the crappy design part).

Step 5:
If the pull cord didn't retract, you need to remove the other two pieces (lower black plastic piece and metal disc with ribbon). The metal ribbon has a nub on the outside that hooks onto a plastic piece in the bottom of the orange plastic housing. You'll need to wind it around once or twice to "charge" the ribbon, then wind the string back on the black plastic piece. The black piece should slide into the center of the metal disc and catch the inside of the ribbon. Once you get these two together, again, make sure the metal ribbon piece is wound once past normal and then you have to slide both parts together down into the housing catching the metal ribbon loop over the orange plastic nub. It's tricky, it took me a few tries. Once you get that you can try and test the pull cord, you'll see it pull out and feel it retract due to that metal ribbon. Then go back and do step 4, where you put the upper black plastic piece with spring back into place and replace the little metal guardrails.

This whole thing took a few tries, and the pull cord still gets stuck out. It's a major pain to say the least, but after the 5th time fixing it, you'll be a real pro.