If this poorly written paper confuses you, here are the key points:
Suppose you heat the hot side of a Stirling engine using a combustion process. Then if you assume that the flame temperature Tf is the high temperature Th in the Carnot efficiency (Th-Tc)/Th, boy, are you going to be disappointed.
The picture the paper uses is the following: The heated air comes out of the combustion at the flame temperature Tf. Then it is ducted to the wall of the Stirling engine, which it heats to a temperature Th (the one in the Carnot formula). This cools the air down to Th. Then this still very hot air flows away as exhaust gas, and the remaining heat in it is lost. Well, it will heat the room in which the engine is located. Due to this loss of heat in the exhaust gas, the real efficiency is less than Carnot: Carnot only tells you how much of the heat entering the engine is turned into work. If you lose heat to the room from the exhaust gas without getting any work from it, that produces an additional loss in real efficiency.
Now assume that the high engine temperature Th is close to the flame temperature Tf. In that case the hot air hardly cools down going from temperature Tf to Th, so hardly any of the heat in the hot air goes into the engine; almost all of the heat in it goes directly into the room. So the real efficiency is virtually zero.
On the other hand, if you take almost all the combustion heat out of the air, you leave the air at a Th which is almost the ambient temperature Tc. So now almost all the heat goes into the engine. Unfortunately, since now the high engine temperature Th is almost the same as the ambient cold temperature Tc, now the Carnot efficiency is almost zero. So almost none of all that heat now going into the engine is turned into work. So once more the efficiency is almost zero.
The best you can do is operate somewhere in between, where Th is well less than Tf to send a lot of the available heat into the engine, but also a lot higher than Tc, so that the Carnot efficiency does not become horrible. The geometric average of Tf and Tc seems like a good idea, does it not?