The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram.
For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 & w_1 \\ v_2 & w_2 \\ \end{array} and so forth for a $3$ or more -dimensional parallelogram.
How is possible to visualize that, or understand that intuitively?
You can see the determinant when changing variables of integration in two or three dimensions. The determinant can be viewed as the Jacobian of the transformation
$$ (x,y) \mapsto (v_1x+v_2y,w_1x+w_2y)$$
and this maps the unit square onto the parallelogram spanned by $(v_1,v_2)$ and $(w_1,w_2)$. The unit square has area 1 and therefore the determinant balances the equation.