This is true.
Consider the image of a RAV4 hybrid instrument panel:
In it, on the left, you see that close to 50% of the maximum load, you are in the ECO region and between 50-100% of the maximum load, you are in the PWR region (the upper end of the PWR region is accompanied by loud sounds coming from a high-revving engine).
Or consider the basic engine operating line of a Toyota Prius engine (the contours are g/kWh contours):
You can see that max efficiency is at about 2100 RPM and 100 Nm, whereas maximum power is 5500 RPM and 130 Nm. Max efficiency occurs at 2100*100/(5500*130) = 0.29 times the maximum load. Or, approximately 30%.
Now, why is the ECO region in the RAV4 hybrid up to 50% of max load and not up to 30% of max load? This is explained by the electric boost inherent in hybrids (it provides the 20% gap).
If you don't have a hybrid, then ideally, you should be using 30% of the maximum power of the engine. This means nearly but not completely full throttle, and 2000-2500 RPM. A smaller engine needs to be operated at closer to full throttle and at higher RPMs, meaning lower gears. This wastes fuel.
A good example: Toyota Yaris switched from 1.33 litre engine to 1.5 litre and the emissions and fuel consumption went down. The larger engine indeed was more economical!
Another good example: Toyota Prius used to be 1.5 litre but nowadays is 1.8 litre. Fuel consumption went down after switching to a larger engine.
Of course, this applies only to reasonably sized engines. If you want a 5 litre V8 in a passenger car, you shouldn't be expecting low consumption.