DIAGNOSIS The slide plate is seized with carbonized grease and gunk. Heat cycles bake everything onto the metal rails.
FIX Don't force it, you'll bend the handle. Fire the grill up on high (450°F+) for about 30 minutes to get everything hot and expand the metal. Using heavy-duty grill gloves, try to work the slide back and forth. Once you get some movement, shut it down, let it cool, and pull the plate out completely for a deep scrub with a grill brush and degreaser.
IF THAT DIDN'T WORK The linkage rod that connects the handle to the plate might have come disconnected inside the grill. You'll need to remove the heat deflector and drip tray to look inside and see if the rod is still attached to the plate's bracket.
HEADS UP After every high-heat or greasy cook where you use the sear plate, move it a few times while it's still hot to keep it from sticking. Prevention is way easier than the fix.