Why is the random variable of this martingale integrable? More specifically, how can I see that $E|S_n|<\infty$?
We begin by describing three examples related to random walk. Let $\xi_{1}, \xi_{2}, \ldots$ be independent and identically distributed. Let $S_{n}=S_{0}+\xi_{1}+$ $\cdots+\xi_{n}$ where $S_{0}$ is a constant. Let $\mathcal{F}_{n}=\sigma\left(\xi_{1}, \ldots, \xi_{n}\right)$ for $n \geq 1$ and take $\mathcal{F}_{0}=\{\emptyset, \Omega\}$.
Example 4.2.1. Linear martingale. If $\mu=E \xi_{i}=0$ then $S_{n}, n \geq 0$, is a martingale with respect to $\mathcal{F}_{n}$.
To prove this, we observe that $S_{n} \in \mathcal{F}_{n}, E\left|S_{n}\right|<\infty$, and $\xi_{n+1}$ is independent of $\mathcal{F}_{n}$, so using the linearity of conditional expectation, (4.1.1), and Example 4.1.4, $$ E\left(S_{n+1} \mid \mathcal{F}_{n}\right)=E\left(S_{n} \mid \mathcal{F}_{n}\right)+E\left(\xi_{n+1} \mid \mathcal{F}_{n}\right)=S_{n}+E \xi_{n+1}=S_{n} $$
Recall, from measure theory, that a random variable $X$ admits an expectation $E(X)<\infty$ if and only if $E|X| < \infty$.
An application of the triangle inequality then tells us that $$E|S_n| \leq E|S_0| + E|\xi_1| + \ldots + E|\xi_n| < \infty$$