A radiator's core is made up of fins and tubes. These are designed to afford the maximum surface area in the smallest space.
The circuitous route followed by the coolant within the radiator allows for heat to be drawn out to the radiator walls.
Modern car radiators have special sensors that measure the temperature of the coolant after it has passed through the radiator. This measurement is transferred to a gauge fitted on the dashboard that the driver of the vehicle can monitor.
The transmission cooler is inside the radiator tank. Just like the radiator itself, its job is to exchange energy.
The oil from the transmission enters the cooler and its heat transfers to the coolant in the radiator surrounding it.