What does it mean that $X_{t}$ is "observable" at time $t$ in the context of an adapted process?
My guess is that it is possible to determine which set the $\omega$ which is "under development" belongs to(and not belong to), which in turn imply a certain value for $X_{t}$, at time $t$.
I seen this formulation or jargon used in several sourses.
A filtration $\{\mathcal{F}_{t}\}_{t \in [0,T]}$ is a collection of $\sigma$-algebras which satsify $\mathcal{F}_{s} \subseteq \mathcal{F}_{t}$ if $s \leq t$. The intuition of $\mathcal{F}_{t}$ is the information available at and up to time $t$. What is information? It is the set of "events" which you can say "yes" or "no" as far as whether or not it occurred. So $A \in \mathcal{F}_{t}$ means we can say at time $t$ whether the event $A$ occurred or not.
It's natural that if you know information at time $s$, you know that information at any later time $t$, which is why $\mathcal{F}_{s} \subseteq \mathcal{F}_{t}$, which in words says "the information you know at time $s$, you know at time $t$" (as long as $s \leq t$). If $A \in \mathcal{F}_{s}$, then $A \in \mathcal{F}_{t}$, i.e., if we can say "yes" or "no" to whether the event $A$ occurred at time $s$, then we can certainly answer the same question by the time $t$.
A process $X_{t}$ is adapted to a filtration if the random variable $X_{t}$ is $\mathcal{F}_{t}$ measurable for each time $t$. What does this mean? It means for each Borel set $B$, $X_{t}^{-1}(B) \in \mathcal{F}_{t}$, where $X_{t}^{-1}(B)$ is the preimage of the set $B$.
What is the interpretation here? Well, $X_{t}^{-1}(B)$ is the set of possible outcomes $\omega$ such that $X_{t}(\omega) \in B$. So $X_{t}^{-1}(B)$ is all of the possible outcomes that cause $X_{t}$ to be in $B$. So you can read $X_{t}^{-1}(B)$ is the event that "$X_{t}$ is in $B$".
That $X_{t}^{-1}(B) \in \mathcal{F}_{t}$ means the "information" of whether or not $X_{t}$ is in $B$ is contained in all of the information known at the $t$, since $\mathcal{F}_{t}$ is all of the information you know at time $t$. That means at time $t$, you know whether or not $X_{t}$ is in $B$, since you know whether or not the event $X_{t}^{-1}(B)$ occurred (and you know whether or not it occurred at time $t$ precisely because $X_{t}^{-1}(B) \in \mathcal{F}_{t}$).
So, all of the above explanation to say this: If you have a set of information for each time $t$, and the information you know at time $s$ you know at the later time $t$ (so that the "informations" form a filtration), then a process $X_{t}$ is adapted to a filtration if, for each Borel set $B$ (i.e., each "pocket"/"set" in $\Bbb R$), you know if $X_{t}$ is in $B$ at time $t$. Note that for each $x \in \Bbb R$, $\{x \}$ is a Borel set. That means at time $t$, we know if $X_{t} \in \{x \}$. So that means we know the value of $X_{t}$ at time $t$, since the value will be the $x \in \Bbb R$ such that $X_{t}^{-1}(\{x\})$ occurred.
SO, $X_{t}$ is adapted with respect to the filtration ("information") $\{\mathcal{F}_{t}\}$ if at each time $t$, the information available at that time, which is represented by $\mathcal{F}_{t}$, tells us the value of the process at time $t$, i.e., the value of $X_{t}$. Short version: a process $X_{t}$ is adapted with respect to a set of information if you know the value of $X_{t}$ from the information that you know at time $t$.
Let $(\Omega,{\cal F},P)$ be a probability space and let $({\cal F}_t)_{t=0}^{+\infty}$ be a filtration, that is, a sequence of $\sigma$-algebras such that ${\cal F}_0\subseteq{\cal F}_1\subseteq\ldots\subseteq{\cal F}_t\subseteq\ldots\subseteq{\cal F}$. The process $(X_t)_{t=0}^{+\infty}$ is adapted if $X_t$ is ${\cal F}_t$-measurable for all $t$.
The $\sigma$-algebra ${\cal F}_t$ describes the information that you have at time $t$. The process $(X_t)_{t=0}^{+\infty}$ is adapted if you can observe it at time $t$, that is, if all its level sets $\{\omega\,|\,X_t(\omega)\leq c\}$ are contained in the information at time $t$.
