Its an IDC connector as I and NickZA noted, not a crimped connector.
The green shell fits with the square holes over the contacts, down to the board, and the open side facing the pin name markings. The wires press flat into the open side of the shell, and into the open sides of the contacts and the narrow slots in the contacts pierce the insulation and make contact with the cores. The insulation is held by the little fingers at the tip of the green shell. Its intended to be assembled once, with some sort of parallel jaw tooling with a shaped anvil to support the individual wires while pushing them home, probably before soldering to the board, and is intended to be non-removable.
Disconnect the green connector on the other end of the wire from the white shelled right-angle header on the display board so you can work on the wire to button board connection without worrying about protecting the fragile display. Wiggle it out gently - don't rip it off!
You *may* be able to reassemble it by putting the shell back in place then clamping that end of the board in a vice so the shell is supported by the top of the vice jaw, and pressing each wire back into place (after trimming the wire end to remove any section with damaged insulation), using some sort of flat ended metal tool not much thinner than the slots in the shell, but if the contacts have been bent or stretched even slightly, you may not get a reliable contact. Check each connection with a continuity tester before you plug it back into the display board. If you cant get a reliable connection on all the wires, you'll have to desolder the contacts, and solder the wires through the same holes direct to the board, discarding the contacts and shell.