Why Constant Returns to Scale Suggests Certain Relation
The result follows from Euler's Theorem on homogenous equations. This theorem states that if a function $f(x,y)$ is homogeneous of degree $\lambda$ then the following holds:

$\lambda f(x,y)= x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$

A (production) function with constant returns to scale is homogeneous of degree 1 by definition. The definition of constant returns to scale is basically the same as the definition of homogeneity of degree 1. That means $\lambda=1$, which proves the result you need.