As the Sun ages, what will happen to life here on Earth? Why?

Our Sun will slowly get hotter as more and more Helium accumulates in the Sun's core. In ~1 billion years, liquid water will no longer be possible on Earth's surface, and the oceans will completely boil away, resulting in a Venus-like planet. Only thermophyllic bacteria may be able to survive, perhaps at the poles. In ~4-5billion years, our Sun will have converted ~10% of its Hydrogen to Helium, and thus become so hot that Helium to Carbon fusion will begin in earnest, causing the "Helium flash" which will convert our Sun into a red giant, and swallow and vaporize all of the rocky inner planets. It may even blow off a planetary nebula. (Contrary to what the others have said, our Sun will not 'run out' of Hydrogen.)