Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.
What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $S$ is a topological embedding from what we proved so far.
Up to the point you have highlighted, we have shown that for any point $p$ in $S$, we can define a chart $(\psi,V)$, where $V$ is open in $S$ and contains $p$, with $\psi\colon V\to \widehat V\subset \Bbb R^k$ a homeomorphism. The inclusion map $\iota\colon S\hookrightarrow M$ is a topological embedding because $S$ is given the subspace topology, hence $\iota$ is a homeomorphism onto its image.