After the screen is removed you'll see this screw that holds the dpad support. Remove it.
Lift up the dpad support, but leave the L button cable in place.
Remove the screw circled in red. Circled in green is the analog stick cable connector. It has a clasp on it that has to be carefully lifted up to release the cable. This clasp is very easy to break off!
This is the joystick removed from the PSP 3000. Installation is the reverse of removal.