How can a red giant grow so big? Supposedly when the sun becomes a red giant, it will grow big enough to swallow Earth.
However, this requires the radius of the sun to expand by a factor of approximately 215×, meaning its volume would have to expand 10,000,000×.
Maybe it's just me, but something about this doesn't feel intuitive :-) especially given that the sun isn't gaining much mass. Exactly what about fusing elements heavier than hydrogen implies that the reactants and/or products take up 10,000,000× as much volume? Nobody ever explains this when explaining the stellar lifetime, and I don't understand why it should be the case. (In fact, I would have expected fusion to result in a decrease in volume, given that nuclei are combining...)
Edit:
It seems there are 2 types of red giants, some that occur during hydrogen fusion, some helium.
If the answer is different for these two types, I'd at least like to know the answer for the helium type (though of course I'd appreciate one that goes further and addresses both).
In my mind, none of these explanations really cover the actual reason that red giants expand.  Indeed, this subject seems like an area where people just make up anything that sounds plausible, but it's often quite wrong (Fraser Cain mentions both light pressure and a higher volume in the fusing shell, but light pressure plays no role at all, and the volume of the shell isn't much different than the core, which is way smaller than the Sun's core). So let's get the story straight.

Many of the descriptions do include some of the key elements, including the fact that you have a shell of hydrogen fusion going on atop an inert degenerate helium core.  But the key reason for the expansion is that the way this shell self-regulates its fusion rate is quite a bit different from the way the Sun's core is self-regulating its fusion now.

Now, the Sun's core self-regulates its fusion rate to match the rate that energy (in the form of light) is diffusing out through the mass of the Sun.  The way it does this is essentially the same in all main-sequence stars: they regulate their core temperature, which is why the core temperature of more luminous main-sequence stars is a little higher.  But this is not at all the way the shell fusing in a red giant self-regulates its fusion rate-- it cannot regulate its temperature, because the temperature is handed to it by the gravity of the degenerate core its sits upon.  (This sets the temperature via the virial theorem, that is the key way the degenerate core affects the shell-- it sets its temperature.)  Since the shell doesn't regulate its own temperature, the temperature tends to be quite high, especially as the core gains mass (this is why the luminosity rises with time).  Fusion is very sensitive to temperature, so being stuck with a very high temperature makes the fusion rate go berserk.  The rest of the star cannot support this spectacular fusion rate, so something else happens.

The star puffs out, and in so doing, we find the way the shell regulates its own fusion rate: it lifts weight off the shell.  That reduces the pressure in the shell, which compensates for the high temperature and brings the fusion rate down to what the rest of the star can manage (which is set by the rate light can diffuse through the shell).  So there's the real reason-- the star must find a way to lift weight off its crazy-high-temperature shell to keep the fusion rate from going nuts, but the upshot is that the fusion rate is still pretty high, and gets higher as the mass of the core rises, forcing the shell temperature up ever more and forcing the star to puff out even more.