Click on the pictures to enlarge.

Check the kit contents. You should have a board, small flex cable, USB cable, and tri-wing screwdriver. You'll also need a small Phillips screwdriver and carving tool (a Dremel tool is strongly recommended, though you might get by with a razor blade, x-acto knife, etc.)
Remove screws and open the DS plastic shell.
Unscrew and remove the stylus holder piece from inside the DS shell. Cut the marked pieces from the shell.
Continue cutting a hole for the USB connector, using the DS Capture board to check the hole placement. Carve inward around the hole so the USB connector will sit flush with the outside.
Check the fitting of the board inside a closed DS shell. Don't remove the adhesive backing and stick the board down yet, just tape it down for now. Putting the shell back on can be tricky; it helps to angle it in, starting from the side of the USB connector. You should also seal the stylus hole shut to keep dirt and foreign objects from getting inside.
The remaining steps will be easier if you remove the right shoulder button first (mind the spring!).

Disconnect the upper LCD cable from the DS mainboard by carefully flipping up the lock (the small black tab) on the connector and pulling the cable out. The locking tabs are delicate and can break easily, use extra care when opening and closing them.

Insert the small flex cable into the LCD connector, gold contacts facing down, from the cable edge with longer gold contacts. While the cable is fully inserted, lock it in place by flipping down the black tab.
Fully insert the other end of the small cable into the top connector (labeled P3) of the capture board and lock it in place by flipping the plastic tab down.

Insert the LCD cable (from the top display) into the connector labeled P2 on the capture board and lock it in place.
Before finishing up, test the board to make sure it's installed correctly. Connect the battery (facing the right way!) and turn on your DS. Verify that both screens look normal.

If everything checks out ok, remove the adhesive backing and firmly stick the board into place. Use the white marks on the board to line it up with the metal shield underneath. Put the shoulder button + spring and outer shell back on.