Consider two models of the Earth/Sun/stars system. Model (A) says the Earth is at the center and the Sun and stars revolve around the Earth. Model (B) says that the Earth revolves around the Sun and the stars are background objects.
Model (A) requires no relationship between the apparent rotation rate of the Sun and that of the stars. On the other hand Model (B) does. To see what that relationship is, take a look at these diagrams.
Solar Noon (day 1): The observer on
Earth sees the sun crossing the North-South line in the sky.
One sidereal day later: The Earth has
rotated once as seen from the distant stars. It has also revolved, just a bit,
around the sun.
Solar Noon (day
2): This happens about 4 minutes later because the Earth-Sun line has moved.
The result is, the ratio of one solar day to one sidereal day is a bit greater than one. How much more? The math's a bit complicated, but it works out to just what is observed.
Model (B) explains the ratio, Model (A) doesn't. So this is evidence for Model (B).