I am reading the notes, trying to understand the proof of Proposition 3.5:
I have no idea why $X$ is homeomorphic to $S^2\times S^1$ with the interior 3-ball removed. I think by the "obvious product structure on B", $B=D^2\times [0,1]$, with $D^2\times\{0\}$ and $D^2\times\{1\}$ identified to be $B\cap(S^2\times\{\pm\epsilon\})$ respectively, but I have no clue why it gives "$S^2\times S^1$ with the interior 3-ball removed".
Also, I wonder how to see $\partial X$ is a separating 2-sphere.
You're right about the obvious product structure on $B$, in fact calling it a 3-ball is kind of irrelevant.
The product structure you describe is the real point: I think it also helps to understand why "$X$ is $S^2 \times S^1$ with the interior of a closed 3-ball removed." Here's some details
First, let's be more specific and chooose an embedded disc $\Delta \subset S^2$ so that $$B \cap (S^2 \times \{-\epsilon\}) = \Delta \times \{-\epsilon\} $$ and $$B \cap (S^2 \times \{\epsilon\}) = \Delta \times \{\epsilon\} $$ Now one has to argue that $X$ can be given the form $S^2 \times [-\epsilon,+\epsilon] \sqcup D \times [0,1]$ modulo identifications $$\Delta \times \{+\epsilon\} \approx D \times \{0\} \qquad D \times \{1\} \approx \Delta \times \{-\epsilon\} $$ Also, under this homeomorphism, letting $\pi \in \Delta$ be the center and $p \in D$ be the center, one has to argue that $\gamma$ has the form $$\bigl(\{\pi\} \times [-\epsilon,+\epsilon]\bigr) \sqcup \bigl(\{p\} \times [0,1]\bigr) $$ modulo identifications $(\pi,+\epsilon) \approx (p,0)$ and $(p,1) \approx (\pi,-\epsilon)$.
Now write $S^1$ as a union of two semicircles denoted $\alpha,\beta$ meeting at their endpoints, replace $[-\epsilon,+\epsilon]$ by $\alpha$, and replace $[0,1]$ by $\beta$, and perhaps the homeomorphism you desire is now clear.
Regarding your final paragraph, $X$ is a compact 3-manifold with boundary, and its boundary $\partial X$ has been expressed as a union of a disc $(S^2-\Delta) \times \{-\epsilon\}$, an annulus $\partial D^2 \times [0,1]$, and a disc $(S^2 - \Delta) \times \{+\epsilon\}$, hence $\partial X$ is homeomorphic to the $2$-sphere. But $M$ is a connected, oriented 3-manifold. So the closure of $M-X$ is also a compact 3-manifold with boundary $\partial X$. Thus $\partial X$ separates $M$ into two components: $X$ and the closure of $M-X$.