That is a thermally bonded ribbon circuit.
You can remove it from the existing PCB with a hair dryer, though you'll probably destroy the first one you try. Also be careful the hair dryer blast does not hit the plastic housing if you have plans for that - they can warp at surprisingly low temperature.
You can then re-attach the ribbon to a custom PCB or breakout, using something like a solder station with a large tip set to absurdly low temperature (try it against the edge of the ribbon to make sure it doesn't melt it) or possibly a small sealing iron like used for model airplane covering; likely the factory uses some sort of heated bar press.
An LCD display requires an AC voltage between a select and a segment pin. If you look around, you'll probably find some app notes on driving them from an MCU by leveraging the 3-state capability. Essentially you toggle the select and the segments you want, feeding them a square wave in opposite polarity from each other, select in same polarity with the segments you don't want, and assign the other selects as inputs. Then you cycle to the next select and drive the segments that should be on for that, etc. A temporary firmware with serial input can let you probe around the possibilities and find the mapping.
It is possible to do some crude tests with a single flashlight cell and some wires, but with DC excitation the segments will only pulse briefly before fading, and continued application of DC is undesirable as there are electrochemical degradation concerns. Setting up an MCU to pulse two pins in opposition would be a better test source.