When the fragmentation warhead goes off, the missile is basically disassembled from the middle section. The nose section, already ahead of the warhead explosion going on behind it, just keeps right on going. And the fragments themselves are oriented in such a way as they go out perpendicular to the axis of travel of the missile body, so none of them will be heading in the direction of the missile nose section anyway.
Some warheads are filled with thousands of little hardened steel or tungsten sticks, almost like concrete nails, while other designs have this thing called a continuous rod design, which is a bunch of long thin metal stakes, welded at the ends to each other, and with an explosive charge inside. When the charge blows, the stakes expand instantly into a big, wide metal circle, almost like a giant metal hula hoop. This expanding hoop cuts right through the structural parts of an aircraft, like a weed whacker string through grass. It just cuts off whatever it passes through. Wings, fuselage, tail, whatever.
Anti aircraft missiles are fascinating devices. Way more complex than missiles for ground or naval targets.