The determinant measures area
The Shear preset $\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$ slides the top edge of the unit square sideways while the base stays put. What happens to the square's area?
Scale stretches $x$ by $2$ and squeezes $y$ to $0.75$: the stats read $\det A = 1.50$, area $= 1.50$. Shear leans the square into a parallelogram, yet both stats hold at $1.00$ — base and height never changed. Rotate 30° moves every dot on screen and changes nothing: $1.00$ and $1.00$. In every case the two readouts agree exactly.
That agreement is the definition. The determinant of a $2\times2$ matrix is the area of the parallelogram its columns span — the image of the unit square. Because a linear map treats every patch of the plane the same way, the one number is the area scale factor for all regions at once: a region of area $S$ lands with area $\det A \cdot S$. From the entries, it is one product minus the other:
Check it against the presets: the scale gives $2 \cdot 0.75 - 0 = 1.5$, the shear gives $1 \cdot 1 - 1 \cdot 0 = 1$, and the rotation gives $0.866^2 + 0.5^2 = 1$. A shear moves area sideways without creating any; a rotation only carries it around.
About this model
What it represents: the exact area behavior of a $2\times2$ linear map — the image of the unit square is precisely the parallelogram spanned by the columns, and $ad - bc$ is precisely its area factor. Nothing here is approximated.
What it leaves out: higher dimensions, where the determinant scales volume and the formula grows past $ad - bc$ into cofactor expansions — and nonlinear maps, which get only a local version of this story through the Jacobian determinant in multivariable calculus.
Source: Margalit and Rabinoff, Interactive Linear Algebra, Determinants and Volumes (Georgia Tech, free online) — the full statement, including the $n$-dimensional case.
Compute the determinant of the Scale preset, $\begin{pmatrix}2 & 0\\ 0 & 0.75\end{pmatrix}$.
Rotate 30° moves every point in the plane except the origin, yet its determinant is $1$. What is that $1$ actually claiming?
Transfer: a matrix $B$ has $\det B = 3$. A triangle of area $5$ is fed through $B$. What is the area of its image?
The determinant is the area scale factor of a linear map. $\det A = ad - bc$ is the area of the parallelogram the columns span, and every region in the plane has its area multiplied by that same number. One number summarizes what the whole map does to area.