Basically in a nutshell you do this. Remove that little toasted board. Using your soldering iron and solder sucker (or desolder braid, or desolderer etc) remove all of the solder on the back of the toasted pins on the board. After that the old pins should pull out with some pliers. Put the new pins through and resolder. Make sure you clip off the one that was missing from the original, in your pic its third from the left. Its missing for a reason. Your going to put your little plug in that position in the connector later.
Then take that crusty connector and snip one wire off at a time. You strip and expose maybe .25" from the end of the wire. Rest it in the crimp contact. The goal is that the innner little collar should wrap around the exposed wire, but you don't want to have the wire stick out much further from that crimp, you don't want a lot of exposed wire after that inner crimp and you don't want it touching the part of the crimp contact that curves up. The outer collar of the crimp contact gets crimped second and the goal is that it should be crimping around the insulation at the end of the wire. You really want three hands for this, they are so tiny and you want to be as precise as possible. Check the first link above that's for pin repair, in section four its shows what you want.
Then you slide the crimp into your new connector, it should click into place and if you give it a little tug it should not come out. There is a little tab on the top of the crimp that locks into that little cut out in the connector housing. If you do need to remove one, take a tiny eyeglass screwdriver and push that little tab exposed through the cutout to release it. You do each wire one at a time so you make sure they are staying in order. Take a picture beforehand too and compare before you plug back in and turn on. Make sure you have the orientation of the connector the correct way when you start or you can find out that you just did it backwards, IE all of the wires are now in reverse order. I've done that.
The great thing is, is you mess up a crimp, just cut it off and try again, that's why you bought a ton of them. If you plug it back in and it doesn't work 100% one of the crimps was not done correctly, no harm no foul, just figure out which one is loose and try again. Don't forget to put a plug in that connector where there is no pin