The intuitive way to think about it is to understand that there are multiple changes that, in essence, amplify each other. Amplification in astronomy isn't all that uncommon. It explains why gravity can make massive objects so small, because as the massive object gets smaller, the gravitation and weight of the object grows exponentially. In a sense, the opposite happens with a Red giant. The gravity at the surface grows low enough that the star kind of enters a run-away expansion.
The expansion of the star late in it's life is exponential. That's why it can expand so much.
If the sun was to double in size but it's mass was to remain unchanged. In this hypothetical, the new Sun's surface gravity is divided by 4. It's escape velocity is divided by the square root of 2, so the outer layer has much less weight, but the escape velocity still binds it to the star. Everything being equal, expanding the sun should cause it to cool, but using the root mean square rule for thermal velocity, if the temperature is divided by 2, the velocity of the hydrogen and helium molecules is divided by the square root of 2.
In this theoretical, they hydrogen atoms on the surface are moving slightly slower, but with 1/4 the gravity, they're more free and they can move further from the star based on their thermal velocity.
If we keep expanding the sun, there comes a point where the outer hydrogen becomes incredibly loosely bound. At red-giant size, say, 1 AU in radius or 215 current solar radii, the gravity is some 46,000 times lower and the hydrogen on the surface only experiences 0.006 m/s^2 gravitational acceleration, but those same hydrogen molecules at red giant temperature (some 3,000 degrees K), are moving about 5.5 km/s. They can fly away from the surface for over a million km based on their thermal energy alone, compared to about 100 km at the surface of the sun currently (based on just under 8 km/s).
In both cases, the outer layer of hydrogen and helium are in equilibrium, it's just that the gravity and red-giant size is so so so much lower that with the red-giant, equilibrium is this very loosely bound very disburse hot gas. But that's only part of the reason.
Consider what else happens as the sun grows older.
The core, where the fusion happens is a comparatively small region in the center. Wrapped around the core is the radiative zone and the conductive zone. which help keep the heat from fusion trapped inside the sun. As a result, over time, the inside of the sun grows hotter and as it grows hotter the core grows larger and it encompasses more and more of the radiative zone.
If we think of the radiative zone as a kind of blanket that traps heat inside the sun, as the core grows larger and more massive, the radiative zone is both stretched and it loses mass to the core, so it becomes thinner in two ways. If the size of the core is doubled, the photons from the core have to travel through 1/4th as many molecules. As the sun gets old enough and most of the fusion happens on the outer edge of the core, there's significantly less of a blanket to keep the heat trapped in. It's not so much that more energy is being created, it's that that energy has an easier path to the outer region of the sun. So you have an amplification effect, as the sun grows larger, the surface gravity drops by the square of the radius and the internal heat has less material to pass through to reach the outer layers, creating an outward pressure as well as delivering more heat.
The internal core collapse can play a role too. Even as the inner core runs out of hydrogen to fuse and it begins to collapse, the act of collapsing generates significant heat.
Not sure that's clear, but that's my attempt to explain what happens intuitively.