Because our world is (locally) Euclidean.
In an Euclidean world straight lines are favored and we subconsciously defined things like distance, area and volume using straight lines. If the real object we are studding is not made of straight lines, we cut it into small pieces, until we get approximately straight lines and then we sum up these small pieces to get the “real” value.
In an Euclidean world there is a direct correlation between Cartesian coordinates and lengths in the real world. For example, a circle in Cartesian coordinates is a circle in the real world; but a circle in the real world is a rectangle in polar coordinates.
Consider a surface bounded by the following curves in our world:
$y=a_0 /x$, $y=a_1 /x$, $y=b_0 x$, $y=b_1 x$
In a world where their vertical lines correspond to a curve defined as $u=xy$ in ours and horizontal lines correspond to the curves $v=y/x$, this complex figure becomes simply a square with sides $a_1-a_0$ and $b_1-b_0$ and area $A=(a_1-a_0)(b_1-b_0) \, u.a$.
In our world it is not that simple and the area bounded by those lines is definitely not that. Now the Jacobian is the tool we use to convert the value of a measurement from one coordinate system to the value that would be obtained if the measurement were performed in Cartesian coordinates. It represents the infinitesimal relation between lengths of an object when drawing in one system to the other. Infinitesimal lengths can always be considered straight lines.
It is important to clarify that Cartesian coordinates is not the only “straight” coordinate system. Any system formed by non orthogonal straight lines can be considered. The thing is that we would consider those systems as sheered and then not necessarily natural.
PS: The Jacobian is always there. We just define $|J|=1$ for a Cartesian coordinate system because the results from this system have direct relation with our world. Some alien species living in a world with a very strong gravitational field (next to a black hole for example ;-), would have defined their $|J|=1$ to a different system.
$$I_c= \int \int dx \, dy$$
As the double integral in Cartesian coordinates, and
$$I_p= \int \int dϱ \, dθ$$
As the double integral in polar coordinates.
Because of the correspondence of Cartesian coordinates and the “real world”, one could notice, either experimentally or by logic (using methods like squaring the circle and cutting the circle) that $I_c$ gives the same face value as the area bounded by the limiting curves. Therefore, one can define the area as
$$A= \int \int \, dA=\int \int dx \, dy$$
And the element of area
$$dA=dx \, dy$$
However, $I_p$ gives a different value (in this case the perimeter of the circle).
From this point onwards, 2 things can be done;
1. Define $│J│=1$ in the starting system
With this definition, applying the Jacobian when passing from the original system to the next will allow you to get the same numeric value, but double integrals will have all sort of meaning, depending on the meaning it has on the starting system, making it a little ambiguous at minimum.
2. Define $│J│=1$ in the Cartesian system
With this definition, all systems will have not only the same numeric value, but all double integrals will result in area of the surface bounded by the curves.
The same for volume integrals.