There are few steps.

The first one. A simply connected homology sphere is a homotopy sphere actually. It follows from the combination of the Whitehead and Hurewicz theorems. By the Hurewicz theorem, $\pi_n(X) \cong H_n(X) \cong \mathbb Z$. Therefore, there is a map inducing homology isomorphism. And by the Whitehead theorem it is a homotopy equivalence.

The second one. Any homotopy sphere is actually topological sphere. It follows from classification theorem in dimension 2. It follows from Poincare conjecture in dimension 3. And finally, in dimension $\geqslant 4$ it follows from the $h$-cobordism theorem.

There are few steps.

The first one. A simply connected homology sphere is a homotopy sphere actually. It follows from the combination of the Whitehead and Hurewicz theorems. By the Hurewicz theorem, $\pi_n(X) \cong H_n(X) \cong \mathbb Z$. Therefore, there is a map inducing homology isomorphism. And by the Whitehead theorem it is a homotopy equivalence.

The second one. Any homotopy sphere is actually topological sphere. It follows from classification theorem in dimension 2. It follows from Poincare conjecture in dimension 3. And finally, in dimension n $\geqslant 4$ it follows from the generalized Poincaré conjecture, proved by Smale 1960-61 for n ≥ 5 and ca. 1982 by Michael Freedman for n = 4.