Is there a good (non-calculational) reason for the formula $|v \times w|^2 + (v \cdot w)^2 = (|v||w|)^2$?

Divide the left side of the equation by $|w|^2$

$$\frac{|v\times w|^2}{|w|^2} + \frac{|v\cdot w|^2}{|w|^2}$$

$v$ crossed with the unit vector in the $w$ direction gives a vector perpendicular to both $v$ and $w$. Taking the cross product again with the unit vector gives us a vector in the same plane as $v$ and $w$, but still perpendicular to $w$ thus

$$\operatorname{Proj}_{w^\perp}v = \frac{w}{|w|} \times \left(v \times \frac{w}{|w|}\right)$$

Similarly, taking dot product gives us that

$$\operatorname{Proj}_{w}v = \frac{w}{|w|} \left(\frac{w}{|w|}\cdot v\right) $$

Since we have chosen a basis $w$ and $w^\perp$ for a vector that lives in a plane, $v$ can be retrieved entirely by vectorially summing these two orthogonal vectors. Thus by Pythogoras we have

$$\frac{|v\times w|^2}{|w|^2} + \frac{|v\cdot w|^2}{|w|^2} = |v|^2$$