How to solder wires to this speaker?
It appears the green / silver object is of the same material as used in making a printed circuit board (PCB).  Which is usually fiberglass with a copper film.  If so it will not bend with out breaking.

Solder two speaker wires to the two silver parts of the PCB.

If you like, try measuring the resistance.  It should not be a dead short, but it will be very close.  The "8 ohm" printed on the back of many common speakers is an impedance as seen from the amplifier which is switching the current back and fourth many times per second.  So it "sees" the impedance of the speaker differently than an ohm meter which is not switching the current back and fourth at all.