A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$? I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true:
$$u \cdot (v \times w) = (u \times v) \cdot w$$
But is there a more high-level reason? I didn't expect the cross and dot product to be connected in this surprising way.

$u\cdot(v\times w)$ is the determinant of the matrix whose columns are $u$, $v$ and $w$. (This can be used as a definition for the cross product).
The identity now follows from the usual determinant rules for permuting rows/columns.
