Let $S^{n-1}$ be the unit $n-1$ sphere centered on the origin. Let $S_p,S_q$ be a small $n-1$ spheres centered on $p,q$ respectively, so small $S_p,S_q$ are disjoint and each is contained in the outside of the other. Let $S_{pq}$ be a large $n-1$ sphere whose inside contains $S_p,S_q$. Choose orientations on all three spheres so that the normal direction points outward from the sphere.
Since each of these spheres is disjoint from the set of zeroes $\{p,q\}$, it follows that the vector field $\bf{v}$ defines continuous functions $f_p : S_p \to S^{n-1}$ and $f_q : S_q \to S^{n-1}$ and $f_{pq} : S_{pq} \to S^{n-1}$. These functions are all just restrictions of the same function $x \mapsto \frac{{\bf{v}}(x)}{|{\bf{v}}(x)|}$ for $x \in \mathbb{R}^n - \{p,q\}$.
By definition, the index of $\bf{v}$ at $p$ equals the degree of the map $f_p$, and the index of $\bf{v}$ at $q$ equals the degree of the map $f_q$.
Next, there is a "boundary theorem" to apply here, although the one you know might not be the same as the one we need. The boundary theorem one needs says that if you have a compact, oriented $n$-manifold with boundary $M$ --- in this case $M$ is the portion of $\mathbb{R}^n$ outside of $S_p$ and $S_q$ and inside of $S_{pq}$ --- and if the boundary $\partial M$ has the induced orientation with normal vector pointing outward from the manifold, and if you have a continuous function from $M$ to $S^{n-1}$ --- in this case, the restriction of the function $x \mapsto \frac{{\bf v}(x)}{|{\bf v}(x)|}$ --- then the sum of the degrees of the maps on the boundary components equals zero. It follows that degree of $f_{pq}$ equals the sum of the degrees of $f_p$ and $f_q$, because we compute the degree on $f_{pq}$ using the orientation induced from $M$ but we compute the degrees of $f_p$ and $f_q$ using the opposite orientation from the one induced from $M$. Thus the degree of $f_{pq}$ equals the sum of the indices of $\bf{v}$ at $p$ and $q$.
Now modify $\bf{v}$ as described to obtain $\bf{w}$. This "modification" is a homotopy from $\bf v$ to $\bf w$, and $S_{pq}$ is disjoint from the support of this homotopy, so the function $f_{pq}$ is unchanged under this homotopy. When the homotopy is complete, the point $r$ is inside the sphere $S_{pq}$, and so, again by definition, the index of $\bf w$ at $r$ is equal to the degree of $f_{pq}$.
Putting it all together, the index of $\bf{w}$ at $r$ equals the sum of the indices of $\bf{v}$ at $p$ and at $q$.
By the way, this point of view on vector fields is well explained in the book of Guilleman and Pollack, which I recommend for more details.
