Why do stars get bigger and brighter when older?
Why does the luminosity increase?
As core hydrogen burning proceeds, the number of mass units per particle in the core increases. i.e. 4 protons plus 4 electrons become 1 helium nucleus plus 2 electrons. But pressure depends on both temperature and the number density of particles. If the number of mass units per particle is $\mu$, then $$ P = \frac{\rho k_B T}{\mu m_u}, \ \ \ \ \ \ \ \ \ (1)$$ where $m_u$ is the atomic mass unit and $\rho$ is the mass density.
As hydrogen burning proceeds then $\mu$ increases from about 0.6 for the initial H/He mixture, towards 4/3 for a pure He core. Thus the pressure would fall unless $\rho T$ increases.
An increase in $\rho T$ naturally leads to an increase in the rate of nuclear fusion (which goes as something like $\rho^2 T^4$ in the Sun) and hence an increase in luminosity.
This is the crude argument used in most basic texts, but there is a better one.
The luminosity of a core burning star, whose energy output is transferred to the surface mainly via radiation (which is the case for the Sun, in which radiative transport dominates over the bulk of its mass) depends only on its mass and composition. It is easy to show, using the virial theorem for hydrostatic equilibrium and the relevant radiative transport equation (e.g. see p.105 of these lecture notes), that $$ L = \frac{\mu^4}{\kappa}M^3,\ \ \ \ \ \ \ \ \ \ (2)$$ where $\kappa$ is the average opacity in the star.
Thus the luminosity of a radiative star does not depend on the energy generation mechanism at all. As $\mu$ increases (and $\kappa$ decreases because of the removal of free electrons) the luminosity must increase.
Why does the radius increase?
Explaining this is a bit trickier and ultimately does depend on the details of the nuclear fusion reactions. Hydrostatic equilibrium and the virial theorem tells us that the central temperature depends on mass, radius and composition as $$T_c \propto \frac{\mu M}{R}$$ Thus for a fixed mass, as $\mu$ increases then the product $T_c R \propto \mu$ must also increase.
Using equation (2) we can see that if the nuclear generation rate and hence luminosity scales as $\rho^2 T_c^{\alpha}$ then if $\alpha$ is large, the central temperature can remain almost constant to provide the increased luminosity and hence $R$ must increase significantly. Thus massive main sequence stars, in which CNO cycle burning dominates and $\alpha>15$, experience a large change in radius during main sequence evolution. In stars like the Sun with $\alpha \sim 4$, the central temperature increases as $\mu$ and $\rho$ increases and the radius goes up, but not by very much.
Hydrogen fusion happens at relatively cool temperatures and low densities compared to the higher order fusion processes. To fuse helium and even heavier atoms, you need much, much more temperature.
Heat dissipation from the star's core to the shell is highly dependent on the temperature gradient. As such, a core at iron producing temperatures looses much more heat to the shell than a hydrogen fusing core does. And because the core looses heat much quicker, it also produces energy much quicker, it's a self regulating fusion reactor, after all.
However, since the shell is so much more effectively heated by the late stage core, and thus much hotter, it's also much less dense. So the total size of the star increases.
TL;DR:
high fusion temperature -> fast heat loss (-> fast energy production) -> high shell temperature -> low shell density -> large size
The size of the star depends on the balance between the gravitational pressure that wants to make it smaller and radiation and thermal pressure from the nuclear reaction that want it to expand. What the nuclear reaction is affects the point at which the balance is achieved.
As stars form and their core gets denser and hotter, the temperature reaches the point when hydrogen is burned into helium. These are the stars of the main sequence.
However, eventually the amount of hydrogen in the core drops, and the energy production from burning hydrogen wanes. The gravity momentarily owerpowers and brings more hydrogen close to the core. This causes a situation when there is no longer a significant amount of hydrogen burnt in the core, but there are regions around the core where hydrogen is still being burnt. Overall, the amount of enrgy produced end up being greater than originally, which means that the star becomes brighter.
However the conservation of gravitational and thermal energy within the star (the radiated energy is much smaller than the total energy in the star, so even with the outflux of radiation the total energy of the star is mostly conserved) implies that when one part of the star contracts, a different part needs to expand. This is called the mirror principle and is predicted by star models. In this case, while the core contracts, the outer shell (the envelope) expands. The result is what is a typical red giant: a helium core, hydrogen burnt in a shell around the core, and a large envelope.
There are other types of red giants (some of them already start burning helium into carbon), but the principle is the same: radiation of the core stops being enough to balance the gravity, gravity collapses the core until new equilibrium is reached (with higher total brightness), energy is transfered to the outer shell, outer shell expands.