How do I install an exterior soffit vent?
It's just friction-fit, probably, if there are no fasteners present. The metal just presses outward with a little spring tension. You'd lift the new on up into the soffit cavity through the gap, then gently pull it down into position, maybe using a short wire hook.

I suppose you could install it upside-down on the surface, but you'd have to use some small screws. They'd need to be pre-drilled to avoid blowing out the edge of the gap. Won't look as nice, either.