Why is $\det \begin{pmatrix} x_1-x_0 & (x_1-x_0)x_1 & \cdots & (x_1-x_0)x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ x_n-x_0 & (x_n-x_0)x_n & \cdots & (x_n-x_0)x_n^{n-1} \end{pmatrix} $
equal to
$$(x_1-x_0) \cdot (x_2-x_0)\cdots(x_n-x_0) \cdot \det\begin{pmatrix} 1 & x_1 & \cdots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & ... & x_n^{n-1} \end{pmatrix}$$
I don't see how we can factor this out.
You extract the factor $x_1-x_0$ from the first line of the matrix, you extract the factor $x_2-x_0$ from the second line and so on. Don't forget that $\det$ is a multilinear map.