what actually opens when it reaches the certain temperature, is it the spring or is it the copper, or does the heat of the engine make the spring softer allowing it to come up and let coolant start circulating?
The spring remains largely the same. There is a capsule of thermoreactive "wax" which heats up and expands, acting against the "closed" spring pressure, and causing the "valve" section to lift allowing coolant flow to the heat exchanger "radiator" to lower coolant temperature.
If it's flowing through the thermostat, it's going to the radiator for heat exchange. If it's bypassed, it is recirculated in the engine to raise operating termperature to an ideal designed setpoint.
The "copper" you mentioned merely serves as the housing to the thermal expansion material, and is usually chosen because of copper's excellent thermal conductivity (around 385 W/m K - better than gold - but less than diamond at 1000 W/m K. But a diamond thermostat capsule would probably be a bit expensive...)
This process is not sudden or binary; the Valve section (yellow hat looking thing in the diagram) may open a little bit, or completely depending on the termperature acting on the thermal capsule.
At some point the coolant temperature is lowered enough to reduce the thermal expansion in the capsule, and the valve seat closes slowly to repeat the process. The hysteresis is natural and fairly unsophisticated. The threshold is tunable, likely with different springs, to provide a 180, 190, 195, etc degree F setpoint depending on the application and design target.
It's a very simple mechanism that has proven completely effective for the last 12 decades or more.