I don't know how they get 70 watts from 12 V @ 0.95 A because that is a tad more than 11 watts.
OTOH, 70 watts sounds about right for a Peltier solid-state cooler with three little fans.
So, I am guessing this thing was designed to plug into a cigarette lighter on an automobile and get 12 V DC power from that. The wall wart power supply and plug you show may be the missing accessory you would need to operate from a convenience outlet.
Here is a 12 VDC power cord that might work.
If this were my POS cooler, I would remove the male connector and install a pair of banana jacks to connect it with a cigarette lighter cable. Or hard-wire in an electrical cord (spliced into the existing wiring) to connect to an external DC power supply. I would also measure the DC current, since it will probably draw more like 6 A than 0.95 A with 12 VDC applied. Pay attention to polarity.
Red is probably positive and
Black is probably negative, but measure the current when you fire it up and see if cool air is coming out to be sure, and also make sure the fans are rotating the right direction.