Understanding a transistor configuration
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See screen dump picture below.



The arrangement is, as you say, known as a "Darlington pair". The aim is to provide much more current gain than can be had with one transistor.
 Current gain overall = Gain_12 = Gain 1 x Gain 2

Typically a transistor will have a Beta (= current gain) of about 100 fo a small signal transistor (can be 50-500+) and say 10-100 for a power transistor (and 1-3 for a few special purpose types. ) So this arrangement may have a Beta of 100 x 100 = 10,000 or perhaps 100 * 10 = 1000.  Either way it means the effective power transistor formed by the pair has MUCH more gain than one normal transistor by itself.
 (Darlingtons have some special features and disadvantages as well which are not too important here).
  Here, using a Darlington pair allows a relatively small opamp output current to control a much larger output current.  The feedback from the lower emitter to opamp- ensures that the output tracks Vr. I won't comment on the rest of the circuit.





Good discussion on Darlington and Sziklai {compound} pairs