This project uses AN OPAMP for its voltage regulation. Its typical DC voltage gain is 100,000 so if the output voltage drops only a tiny amount then it corrects it.
Its output voltage should not drop more than 0.03V when its load is increased from zero to 3A.
Maybe your project has wrong values in the current regulation section. The LED should not light when the voltage is being regulated but it should light to warn you that the current regulation is reducing the output voltage.