Munkres Topology Section 83
First question: Is "obvious deformation" the straight line homotopy
$F(b,t) = \overline{St}(x) \times I \to \overline{St}(x)$?
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{\alpha} \times I$ is continuous"? If not, then what?
I am not asking for proofs but only that to understand this sketch that Munkres has given.
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}\big(tf(b) + (1-t)f(x)\big)$$ where $f:A\to[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $\overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $\overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $\overline{St}(x)\times I$ is coherent with $\{A_\alpha\times I\}$.