In some homework solutions someone posted online, one of the problems was from Hatcher's Algebraic Topology: computing the homology groups of the space which is the union of the boundary of $I \times I$ along with all the points in $I \times I$ with rational first coordinate.
The author of these homework solutions makes the comment
Since $X$ is a subspace of a 2-dimensional manifold, all continuous maps $\sigma: \Delta^k \to X$ have the same boundary as the image of a continuous map $\sigma: \Delta^{k-1} \to X$ for $k \geq 2$. Therefore $H_k(X)=0$ for $k \geq 2$.
Can someone explain to me what this might mean (assuming the author is indeed making sense)? I can't seem to understand.
This is nonsense; I don't know what they're trying to say, and nothing I can imagine they might be saying is a sufficient argument to imply their conclusion. In fact, the statement that $H_k(X)=0$ for $k>n$ if $X$ is a subset of an $n$-manifold is false in general (though it may be true for $n=2$). A famous counterexample is the "$n$-dimensional Hawaiian earring" for $n>1$, which has nontrivial homology in infinitely many dimensions despite being a compact subset of $\mathbb{R}^{n+1}$ (namely, the union of infinitely many shrinking $n$-spheres that all meet at a point). See this paper of Barratt and Milnor (the original reference!) for a proof.