Why exactly is capacitance equal to both Q/V and εA/d?

but what exactly determines this value?

The permittivity of free space is an important factor. It's 8.85418782 × 10-12 farads per metre. I don't think anyone short of a masters in physics can do a proper explanation of why free space has permittivity. I don't intend to start so, all I can say is that it does.

The permittivity of free space can be multiplied by the dielectric constant (relative permittivity) of an insulating material to improve capacitance.

Where did the equation Q/V come from?

Q = CV is just the way things shape up when you look at the units of each quantity (dimensional analysis). It basically follows that the rate of change of charge is current (by definition) and, that results in a rate of change of voltage yielding this modified charge equation: -

$$i = C\dfrac{dv}{dt}$$

This is what EEs mainly use in their day-to-day activities when designing circuits. Basically the formula tells you that a 1 farad capacitor subject to a constant current of 1 amp rises its terminal voltage by 1 volt per second.

It also tells you that if you draw 1 amp from a charged capacitor, the terminal voltage reduces at 1 volt per second for a 1 farad capacitor.

Again all practical usefulness in the subject of EE.

_{But, if you want my personal gut feeling of what capacitance represents in physical units I would say that capacitance is proportional to the thinness and the area of space. Compare with an inductor that, in my opinion, is proportional to the fatness and the volume of space.}

Capacitance is the ability to store the electrical energy in the form of electric field or charge.

So how would we define it mathematically? C = Q/V means how much charge is stores per volt. So if you have two parallel plates at some distance with a dielectric between them then how much charge the plates store for a given voltage will determine the capacitance.

Permittivity is a measure of how much energy you can store in a given chunk of material if you place it in an electric field. The units of permittivity are defined as farads per meter. A farad is defined as 1 coulomb per volt, or units of Q/V. For a capacitor with a plate area A and a plate separation d, εA/d has units of Q/V.

Where capacitance come from?

Let's imagine a metal ball. And we want to charge it. What happens?

Our charge will be acted upon by an electric force directed in the opposite direction to its movement. This means that we will be forced to counteract this force, that is, to do work, to expend energy.

What else will we notice? That each subsequent charge that we transfer to the ball will require more and more energy from us.

Also, if we repeat this experiment with a larger metal ball, we will notice that it is easier to charge (it takes less energy to charge to the same value).

It turns out that the energy we spend on charging depends on the magnitude of the charge on our metal ball. That is, we have some kind of functional relationship between the charge on a metal ball and the voltage (the energy that must be applied to transfer a unit charge).

$$ Q = f(U) $$

This functional relationship is unique for each capacitor. However, for most capacitors, this relationship is linear. Therefore, we can introduce the capacitance as:

$$ Q = CU \Rightarrow C = \frac{Q}{U} $$

However, there are nonlinear capacitors for which this functional dependence is nonlinear. In such cases, we cannot speak of capacitance. Similarly, as for nonlinear resistors, we cannot speak of their resistance. However, as for nonlinear resistors, a differential capacitance is introduced at a particular operating point.

$$ C = \frac{dQ}{dU} $$

The conclusion is that the capacitance of a capacitor is not like the capacitance of, for example, a water bottle. It is like the resistance to charge. A larger capacitance means a smaller resistance to charge. It's just that the resistance is already taken, we are using it for a different physical phenomenon, so we use a different word here.

C = Q/V is the definition of capacitance of an electrode pair. eA/d is the approximate value of the capacitance of two large (relative to separation) parallel plate electrodes. It has no general applicability.