Proof that degree of map is sum of its local degrees (prop 2.30 Hatcher)

By excision, $H_n(S^n,S^n-f^{-1}(y))\approx H_n(\bigcup_i U_i,\bigcup_i U_i-f^{-1}(y))$, and since the open subsets are disjoint, each containing one point of the inverse image, the latter is $\bigoplus_i H_n(U_i,U_i-x_i)$. This second step gives the inclusion maps $k_i$, which in other words each $k_i$ is the induced map for the inclusion map $(U_i,U_i-x_i)\to (S^n,S^n-f^{-1}(y))$.

The inclusion map $(S^n,S^n-f^{-1}(y))\to (S^n,S^n-x_i)$ induces the map $p_i$, and the composition $(U_i,U_i-x_i)\to (S^n,S^n-x_i)$ induces an isomorphism by excision, so $p_i$ is the corresponding projection map.

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Since my own answer confused me a couple years later, here is a more comprehensive diagram showing how it is that $H_n(S^n,S^n-f^{-1}(y))$ is naturally a direct sum $\bigoplus_i H_n(S^n,S^n-x_j)$.

This entire diagram commutes. Every map is marked with how it is obtained. Most are induced by some map of pairs, some of which are marked with "$\approx$ excision" when the excision theorem applies. The map between the direct sums is excision on each component individually. The map $$\bigoplus_j H_n(U_j,U_j-x_j)\to H_n(\bigcup_j U_j,\bigcup_j (U_j-x_j))$$ is induced from all of the inclusion maps $(U_i,U_i-x_i)\to (\bigcup_j U_j,\bigcup_j (U_j-x_j))$ along with the universal property for direct sums, and it is an isomorphism due to the additivity axiom for homology.

The upper part of the diagram shows how going backwards from $H_n(S^n,S^n-x_i)$ through $H_n(U_i,U_i-x_i)$ to $H_n(S^n,S^n-f^{-1}(y))$ via $k_i$ is the natural inclusion for $H_n(S^n,S^n-f^{-1}(y))$ as a direct sum. The lower part of the diagram shows how the induced map $p_i$ is the natural projection for the direct sum thought of as a direct product, using the fact that $H_n$ is functorial. The identification of these maps as the natural inclusions and projections comes from the universal property of direct sums and direct products: $i$ should be thought to vary.