How does one prove/demonstrate that the earth is 4.5 billion years old and that the universe is approximately 14 billion years old?

You can't make the fools shut up - they're not smart enough to understand...
-
Some of the oldest rocks formed (from melted to solid) and enclosed within themselves small bits of Uranium.

Uranium is a radioactive element which decays with known rates, and becomes lead with time (there are different isotopes of Uranium in nature and they each decay to a different isotope of lead, each one with its own time delay). By measuring the mix of Uranium left and lead formed, you get an idea of how long ago the bit of Uranium was trapped in the rock.

Space is expanding and the rate of expansion can be measured. This rate is "linear" (depends on distance) so that it is easy to measure how long a given size of space takes to double in size. By going backwards, we can calculate how much time between now and when the calculations show "almost zero" for size (contrary to popular belief, we cannot go back to exactly zero).

Space expands everywhere, including inside objects. The rate, over very small distances, is very slow (a human body may expand by the size of one atom, over an entire lifetime) so that the normal forces (electromagnetism, gravity...) have no problem holding any object together.

However, there is nothing holding a single photon (particle of light) together so that if a photon exists long enough, it gets stretched by the expansion.

We know the wavelength of photons that are formed under known conditions. We can measure the wavelength of these photons when they finally arrive on Earth. The difference (called cosmological redshift) tells us how long this photon has been traveling through space.

Both methods show that the universe is more than 13 milliard years old (13 billion years, using the American parlance).