What causes the dimensions of a star increase when its hydrogen fuel is exhausted? For example, the Sun is expected to increase its radius 250 times. What causes this if its temperature is expected to fall? How gas can expand if the temperature falls?
The Sun will never run out of Hydrogen. This is a common misconception.
At this moment the Sun is fusing Hydrogen into Helium. This fused Helium remains at the core until it will reach a critical mass. At this point the core will begin to collapse. This collapse increases the temperature and pressure around the core where Hydrogen is being fused causing the increase the Hydrogen fusion which also causes an increase of the radiation pressure making the outer layers to expand and cool.
Once the core reaches a temperature of about 100 million Kelvin Helium fusion starts dramatically (Helium Flash) with high radiation pressure in the core. Here is where the Sun will reach 250 times it's current radius.
When the Helium fusion stabilizes the radiation pressure at the cores decreases making the Sun's radius to shrink.
The temperature will only drop at the surface because as it's expands it's further from the core.