The key reference here is Jean Cerf, "Groupes d’automorphismes et groupes de difféomorphismes des variétés compactes de dimension 3", Bull. Soc. Math. France (1959). The full text is available here.
Let $M$ be a closed 3-manifold, $G$ its group of self-homeomorphisms, $H$ its group of self-diffeomorphisms. (All orientation preserving for convenience.) He proves that $\pi_n(G,H) = 0$ for all $n \geq 0$, thus that the inclusion $H \hookrightarrow G$ is a weak homotopy equivalence - assuming Smale's conjecture (now theorem, due to Hatcher) that the inclusion $SO(4) \hookrightarrow \text{Diff}^+(S^3)$ is a homotopy equivalence. (In fact, he shows that the inclusion is a homotopy equivalents, but the arguments are a bit more delicate.)
For a manifold $M$ with boundary, $H$ and $G$ as above should be the automorphisms restricting to the identity on the boundary. The idea is to show that, if you have a decomposition $M = M_1 \cup M_2$, with $M_1 \cap M_2$ a properly embedded surface, having $H_i$ be $(n-1)$-connected in $G_i$ (for both $i$) is equivalent to having $H$ be $(n-1)$-connected in $G$.
Now we want to induct. Say that a manifold homeomorphic to $D^3$ is order 0; and a manifold that decomposes into order $(n-1)$-pieces is order $n$ if we can't decompose it into pieces smaller than order $(n-1)$. But by Smale's theorem, it's true that $\pi_i(G,H) = 0$ for all $i \geq 0$ when $M = D^3$. Because every 3-manifold has finite order (look at Heegaard decompositions), this proves the theorem.
As for why the third paragraph is true, this is Cerf's "Lemma 0", at the very end of his paper.
Best Answer
By Poincaré duality, $H_2(M)=0$ (use universal coefficient theorem). Applying Hurewicz, we get that $M$ is $2$-connected and that the Hurewicz map $h:\pi_3(M)\to H_3(M)\cong\mathbb{Z}$ is an isomorphism. Letting $f:S^3\to M$ represent a generator of $\pi_3(M)$, we see that $f$ is a homotopy equivalence by Whitehead (using the fact that manifolds are homotopy equivalent to CW complexes).
This is exercise 4.2.15 in Hatcher.