There is no particular need of an inductor in such a circuit.
The description does not include any specifications, but that's a 24VAC CT transformer, apparently with the CT open, so the DC voltage will be around 33V, very close to the maximum input voltage of the 7805 and 7812. Chances are almost 100% it will exceed the absolute maximum peak voltage with the outputs unloaded (inexpensive transformers tend to have considerably higher than rated output voltage when lightly loaded).
The builder has put a silly (unnecessary) heatsink on the bridge rectifier but nothing on the regulators.
The rectifier is good for approximately 1.2A DC output with that transformer, however allowing 600mW on each TO-220 package you can draw only about 20mA from the 5V line and about 29mA from the 12V line before they get very hot.
Sometimes a large inductor (heavy and expensive) is (more like "was") used between the bridge rectifier and the filter capacitor, and such an inductor would allow higher DC current to be drawn without overheating the transformer, and could reduce the voltage under load, however the inductor will be prohibitively expensive. For example, a 2.5H 300mA choke is about $28, and is only good for 300mA.