The [Primer][1] has "model proofs" for various steps of the proof. 
[Bleiler's notes on Casson's lectures][2] are also very good on these topics.  (The notes cover far less material than the Primer, so they are much much shorter.)

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I find the given statement confusing, because the conclusion is not as strong as it should be. I think the conclusion should be as follows:

> If $\gamma$ is a path in $\Sigma$ whose endpoints are fixed by $T$,
> and if $T \circ \gamma$ is homotopic to $\gamma$ (rel endpoints), then there is a homotopy of $\gamma$ (rel endpoints) making $\gamma$ disjoint from the union $\cup C_i$.

**Proof:** We prove the contrapositive. Homotope $\gamma$ (rel endpoints) to have minimal intersection number with $\cup C_i$.  Suppose that the result still meets $\cup C_i$. So $\gamma$ crosses one of the curves, say $C$.  Let $(B_j)_j$ be the subcollection of $(C_i)_i$ which are all parallel to $C$.  Let $X \subset \Sigma$ be an annulus that contains all of the $B_j$ as essential curves in its interior.  We may arrange matters so that $\gamma$ has minimal intersection number with $\partial X$.  Let $\delta$ be a component of $\gamma \cap X$ which meets some of the $B_j$. We may (and do) assume that the endpoints of $\delta$ are fixed by all of the $T_i$. 

Let $\Sigma^C$ be the cover of $\Sigma$ which is homeomorphic to an annulus and where $C$ does not unwrap.  We use the initial point of $\delta$ as our base-point; let $\gamma^C$ be the resulting lift of $\gamma$ to $\Sigma^C$.  Let $\delta^C \subset \gamma^C$ be the resulting lift of $\delta$.  The arc $\delta^C$ crosses the lifts of (the correct) $B_j$ due to our choice of base-point. 

We now lift $T(\gamma)$ using the *same* base-point.  It lifts (up to homotopies in $\Sigma^C$ supported in bigons disjoint from $C$) to a copy of $T_C^n(\gamma^C)$.  Here $n$ is the number of parallel copies of $C$ crossed by $\delta$ (and the sign depends on the sign of the twists along the $B_j$).  Since $T_C^n(\gamma^C)$ has different winding number (than $\gamma^C$) with the curves $B_j$, it follows that they are not homotopic (rel endpoints) in $\Sigma^C$.  Thus (by homotopy lifting) $T(\gamma)$ and $\gamma$ are not homotopic (rel endpoints) in $\Sigma$, as desired.


  [1]: https://press.princeton.edu/books/hardcover/9780691147949/a-primer-on-mapping-class-groups-pms-49
  [2]: https://www.cambridge.org/core/books/automorphisms-of-surfaces-after-nielsen-and-thurston/2AD58B246E36B971CCB92BD5B923BDB9