Multilinearity of the determinant follows from Cavalieri's principle applied to n-dimensional parallelipipeds.
The determinant of a matrix measures the (n-dimensional) volume of the parallelipiped generated by the columns of the matrix:

Multilinearity means that the determinant is a linear function in each column of the input matrix, independently. I.e.:
$$\det \left( \begin{bmatrix}{\color{purple}\lambda} \mathbf{v_1} & \mathbf{v_2} & \dots & \mathbf{v_n}\end{bmatrix} \right) = {\color{purple}\lambda}\det \left(\begin{bmatrix} \mathbf{v_1} & \mathbf{v_2} & \dots & \mathbf{v_n} \end{bmatrix} \right)$$
$$\det \left( \begin{bmatrix} \mathbf{\color{darkgreen} u} + \mathbf{\color{blue} w} & \mathbf{v_2} & \dots & \mathbf{v_n} \end{bmatrix} \right) = \det \left( \begin{bmatrix} \mathbf{\color{darkgreen} u} & \mathbf{v_2} & \dots & \mathbf{v_n} \end{bmatrix} \right) + \det \left( \begin{bmatrix} \mathbf{\color{blue} w} & \mathbf{v_2} & \dots & \mathbf{v_n} \end{bmatrix} \right),$$
and similar formulas must hold for the second, third, etc.. columns.
The first property (pulling out of scalars $\lambda$) is easy to see and already discussed in user2520938's answer. When you linearly scale a parallelipiped in a single direction, you increase its volume by the scaling factor:

To see that the second property holds (multilinearity under addition), translate the two parallelipipeds associated with the right hand side of 2. so that they share a lower dimensional parallelipiped as a common face (the parallelipiped defined by the shared vectors $\mathbf{v_2},\dots, \mathbf{v_n}$). All the slices of this combined object have the same shape, and these slices also have the same shape as the slices of the summed parallelipiped associated with the left hand side of 2. Hence by Cavalieri's principle the parallelipipeds associated with the left and right hand sides of 2. must have the same volume:

For intuition about Cavalieri's principle, just think about a stack of coins. If you take a straight stack of coins and shear it in any pattern, the volume stays the same (image credit for the coin stack to wikipedia):

Of course, the same argument holds when applied to any other column, hence determinant is multilinear in the columns of the input matrix.