Why is the volume of some parallelepipeds defined by the $3$-dimensional vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ given by the following formula?
$$\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}= a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \end{vmatrix}$$
The volume of a parallelepiped of vectors $\vec a=(a_1,a_2,a_3)$, $\vec b=(b_1,b_2,b_3)$, $\vec c=(c_1,c_2,c_3)$ is given as a triple product, like this: $$V=|\vec a\cdot(\vec b\times \vec c)|=\begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{vmatrix}(1)$$
To show that the volume of parallelepiped is equal to determinant given in $(1)$ it is sufficient to show that the volume of parallelepiped is equal to the mixed product of the given vectors.
We will use the fact that $V=BH$, where $B$ is area of base of parallelepiped, or a paralelogram defined by vectors $\vec b$, $\vec c$, on the other hand $H$ is the height of the the parallelepiped. Since $B$ is area of paralelogram defined by vectors $\vec b$, $\vec c$ we have from the formula of area of such paralelogram (proof is similiar to this one) $B=|\vec b \times \vec c|$. So we have $V=|\vec b \times \vec c|H$. To find $H$ we will use the fact that it is a projection of $\vec a$ onto a vector $\vec b \times \vec c$, so that means $H=|proj_{\vec b \times \vec c}\vec a|=|\vec a||\cos\angle(\vec a,\vec b\times\vec c)|$. Keep in mind we are looking at an absolute value of this projection, because if it was negative then the volume will end up being negative, which isn't possible.
From here and from the definition of scalar product we get $V=BH=|\vec a||\vec b \times \vec c||\cos\angle(\vec a,\vec b\times\vec c)|=||\vec a||\vec b \times \vec c|\cos\angle(\vec a,\vec b\times\vec c)|=|\vec a\cdot(\vec b\times \vec c)|$.
This should be as intuitive as it gets. Also if you're wondering why the triple product is equal to determinant you can check here.
It is important to keep in mind we're talking about Euclidean space here.