Opamp I-V convertor working?
When they use the term "very little compliance" they mean that the current from the source will vary significantly if the voltage changes even a small amount.

The circuit you show is a basic Transimpedance Converter; The opamp will drive its output to maintain its input at zero volts by normal negative feedback action.  If we assume that no current flows into the opamp input then all the current from the diode has to flow through the feedback resistor (1Megohm in this case). This means that the output voltage is the current from the source multiplied by the value of the feedback resistor. (Thanks OFRBG).

The effective input resistance will be the feedback resistor divided by the gain of the opamp - since that gain may be one million the input resistance can look like a 1 ohm resistor.  If the effective output resistance of the source is 1 kilohm then 99.9% of the source current will go through the feedback resistor and thus generate the output voltage.

Since the voltage at the input is always zero this means that we can measure the current even from sources with little compliance.

A source with little compliance is equivalent to the source have a low value resistor in parallel - if the voltage across the source changes a small amount the current through the resistor changes and takes away some of the source's output current. In the circuit above the opamp doesn't allow any voltage across this parallel resistance and so no current flows through it.  This ensures that all the current from the source flows into the feedback resistor and generates a voltage that can be measured at the output.

As it happens photodiodes have a lot of compliance at DC and the current will change very little for quite a few volts of reverse bias. However they do have significant capacitance and so their compliance is not good at AC.  Using a transimpedance converter can improve response at high frequencies a great amount and such a circuit is almost always used with photodiodes.

Something else to consider is that the normal TIA requires a compensation capacitor across the feedback resistor to ensure stable operation if there is any significant capacitance at the source (e.g. a photodiode).