This circuit uses an instrumentation or differential amplifier U1 to form a circuit who's output current is controlled by input voltages. The output current is not dependent on the applied load.
The output current is found by:
Differential (instrumentation) amplifiers produce an output described by: gain*(Vin+ - Vin-). See the specific amplifier data sheet to determine how the gain is calculated. Most of these amplifier have a reference input that offsets the output by adding it into the output. This results in: U1out = gain*(Vin+ - Vin-) + ref. Either Vin- or Vin+ can be connected to ground if desired.
In this circuit, Rc is in series with the output of the differential amplifier. The voltage across it is a result of the current flowing through it. This voltage (buffered by U2) is fed back to the Ref input causing the output of U1 to maintain a constant voltage across Rc, and therefore a constant current through it, despite the load applied.
The circuit can safely drive a short circuit and is limited in it's drive ability (source or sink) by the amplifiers output voltage range.