Why does the price and wealth derivatives, weighted by the price and wealth, sum to 0? From Mas-Colell, Whinston and Green (Microeconomic Theory), we have this proposition on page 27. If homogeneity holds,

$$ \Sigma_{k=1}^{L} \frac{\partial x_l (p,w)}{\partial p_k} p_k + \frac{\partial x_l (p,w)}{\partial w} w = 0 $$

That is, the price and wealth derivatives, when weighted by the price, $ p_k$ and wealth $w$, equate to 0. What is the economic logic behind this? Homogeneity implies this -- but for this to hold, we don't require that prices and wealth increase proportionally (am I right?)

From this, we can also derive that the sum of price elasticities (own and cross price elasticities) equate to 0. I understand we can derive this mathematically-- but again, why'd these elasticities equate to 0, economically?

(Also, if these results are held only when prices and wealth increase proportionally, I completely understand -- but the book doesn't make that distinction).
As mentioned by Mas-Colell, Whinston and Green, the equality is true "for all $p$ and $w$". It is a consequence of the budget constraint, which is satisfied for any prices and income values:
$$
\Sigma_{k=1}^{L} p_k x_k (p,w) = w.
$$
As you mention, the equality 
$$ \Sigma_{k=1}^{L} \frac{\partial x_l (p,w)}{\partial p_k} p_k + \frac{\partial x_l (p,w)}{\partial w} w = 0 $$
arises when applying the Euler-theorem to the budget constrained demand functions. As demand functions are homogeneous of degree zero in $(p,w)$, they are left unchanged when prices and income are up- or downscaled by any number $\kappa&gt;0$: 
$$
x_l (\kappa p,\kappa w) = x_l ( p, w) ,
$$
and this equality holds for any value of $p$ and $w$. This implies the former equality and also explains why it is not only true for the upscaled new prices and income $(p,w)=(\kappa p_0,\kappa w_0)$ but actually for any prices and income because both $\kappa$ and $(p_0,w_0)$ can be arbitrary.

What about the economic interpretation? Hmm... I would say that any simultaneous (marginal and arbitrary) change in $p$ and $w$ triggers changes in the values of demands such that the (own- and cross-) price reactions and the income response compensate. It is equivalent saying that all price elasticities and income elasticity add to zero for any demand function $x_l$.