$A \in \mathcal{F}_t$ means: Given the information up to time $t$, we can decide for some fixed $\omega \in \Omega$ whether $\omega \in A$ (or $\omega \in A^c$), i.e. whether the event $A$ happens (or not).
The interpretation for $A \in \mathcal{F}_{\tau}$ is very similar: Given the information up to time $t$ and given that the stopping time $\tau(\omega)$ already occured before time $t$, we want to decide whether $\omega \in A$ (or $\omega \in A^c$).
Example: Let $(B_t)_{t \geq 0}$ be a stochastic process with continuous sample paths and $B_0=0$. Define
$$\tau := \inf\{t>0; B_t \notin (-1,1)\}.$$
- $A:=\{B_{\tau}=1\} \in \mathcal{F}_{\tau}$: Indeed, given the path up to time $t$ and given that the stopping time occured before time $t$, we can easily decide whether $B_{\tau}=1$ or $B_{\tau}=-1$ (since $(B_t)_{t \geq 0}$ has continuous sample paths, these are the only two possibilities).
- $A := \{B_{\tau+1}=0\} \notin \mathcal{F}_{\tau}$: Obviously, this event requires information about the future (compare this with $\{B_{t+1}=0\} \notin \mathcal{F}_t$) and therefore $A \notin \mathcal{F}_{\tau}$.
If $(\mathcal{F}_n)_{n \in \mathbb{N}}$ is the canonical filtration of a stochastic process $(X_n)_{n \in \mathbb{N}}$, then $\mathcal{F}_n$ contains all the information about the process up to time $n$. After observing realizations $X_1(\omega),\ldots,X_n(\omega)$ of the stochastic process, we can decide whether an event $A_n \in \mathcal{F}_n$ has happened, i.e. whether
$$\omega \in A_n \qquad \text{or} \qquad \omega \notin A_n.$$
Since $\{\tau \leq n\} \in \mathcal{F}_n$ this means, in particular, that we can decide whether the stopping has occurred up to time $n$ given the observations $X_1(\omega),\ldots,X_n(\omega)$.
This intuition can be made precise:
Let $(\mathcal{F}_n)_{n \in \mathbb{N}}$ be the canonical filtration of a stochastic process $(X_n)_{n \in \mathbb{N}}$, and let $\tau: \Omega \to \mathbb{N} \cup \{\infty\}$. Then the following statements are equivalent:
- $\tau$ is a stopping time
- If $\omega,\omega' \in \Omega$ are such that $\tau(\omega) \leq k$ and $X_j(\omega) =X_j(\omega')$ for all $j=1,\ldots,k$, then $\tau(\omega') \leq k$.
Summary: $\tau$ is a stopping time if the decision to stop before or at time $n$ (i.e. $\tau(\omega) \leq n$) depends only on $X_1(\omega),\ldots,X_n(\omega)$.
Let's turn to $\mathcal{F}_{\tau}$. Fix observations $X_1(\omega),\ldots,X_n(\omega)$. As we have seen in the first part, we then know whether the stopping has occured up to time $n$, i.e. whether
$$\tau(\omega) \leq n.$$
Suppose for the moment being that the stopping has indeed occured before or at time $n$. Then a set $A \in \mathcal{F}$ is in $\mathcal{F}_{\tau}$ if, and only if, we can decide whether $A$ has occurred (given our observations $X_1(\omega),\ldots,X_n(\omega)$).
Example 1: Let $X_n = \sum_{j=1}^n \xi_j$ for random variables $\xi_j$ which are Gaussian with mean $0$ and variance $1$. Define $$\tau := \inf\{n \in \mathbb{N}; X_n < 0\}.$$ Then the set $$\{X_{\tau} \in B\}$$ is in $\mathcal{F}_{\tau}$ for any Borel set $B$. Indeed: Given that we know that the stopping has occured up to time $n$, we can say which values $X_{\tau}(\omega)$ takes, given the observations $X_1(\omega),\ldots,X_n(\omega)$. In contrast, if the stopping has not occured up to time $n$, the observations $X_1(\omega),\ldots,X_n(\omega)$ don't tell us anything about $X_{\tau}(\omega)$.
Example 2: Let $X_n = \sum_{j=1}^n \xi_j$ for random variables $\xi_j$ such that $\mathbb{P}(\xi_j = 1)= 1/4$ and $\mathbb{P}(\xi_j = -1) = 3/4$. If we define
$$\tau := \inf\{n \in \mathbb{N}; X_n = 100\}$$
then
$$A := \{ \exists k \in \mathbb{N}; X_k =95\} \in \mathcal{F}_{\tau};$$
however, for instance,
$$B := \left\{ \max_{k \geq 0} X_k \leq 100 \right\} \notin \mathcal{F}_{\tau}.$$
Best Answer
The idea is that $F_t$ consists of all events that depend on your stochastic process only up to time $t$. So to say that $\{\omega\in\Omega:\tau(\omega)\leq t\}\in F_t$ means that given any particular outcome $\omega$, to determine whether $\tau(\omega)$ is at most $t$, you only have to look at what happens in $\omega$ up to time $t$.
The idea behind the term "stopping time" is that you are running some stochastic process, and deciding to stop it at some time based on what has happened so far. (For instance, maybe you choose to stop at the first time that your stochastic process reaches some particular value.) An element $\omega\in\Omega$ represents one particular outcome of running the stochastic process, and the time at which you choose to stop for that outcome is $\tau(\omega)$. When deciding whether to stop, you only know what has happened so far (not what will happen later). So, to determine whether you will stop at time $t$ or earlier, you can only look at the outcome of your stochastic process prior to time $t$. In other words, the event $\{\omega\in\Omega:\tau(\omega)\leq t\}$ should be in $F_t$.