sg.symplectic geometry - Clarification of classical field theory lecture notes by P. Deligne and D. Freed

Configuration space is, by definition, the position space of your particles. Phase space, on the other hand, is the space of pairs (position, momentum). The latter has a symplectic structure; the former has a Riemannian structure.

Regarding the relationship between kinetic energy and the Riemannian structure: You will recall from your high school physics class that kinetic energy is $frac{1}{2} mv^2$. Of course the $v^2$ is really the dot product $v cdot v$, in other words it's $g(v,v)$, where $g$ is the Riemannian metric and $v$ is a tangent vector. The $frac{1}{2}$ explains the "twice the kinetic energy" part.