$\mathscr{G}$ is our information in the sense that for all $A \in \mathscr{G}$, we know whether $\omega \in A$.
Let us use the Tickets in a box metaphor, extended to handle $\sigma$-algebras so that the ticket mentions for all $A\in \mathscr{F}$ whether the outcome represented by the ticket belongs to $A$. Now, say that someone else picks the ticket and we don't see it. For any $A \in \mathscr{G}$ we may ask whether the ticket says that the outcome is in $A$ and the person holding the ticket tells us. However, if we ask about some $A \in \mathscr{F} \setminus \mathscr{G}$, we hear "Sorry, you don't know that".
Larger $\sigma$-algebra is more information
This also explains why moving to $\mathscr{G}' \supset \mathscr{G}$ means gaining new information -- now we still get answers to $[X \in A?]$-questions about any$A \in \mathscr{G}$ and additionally to some new questions -- those where $A \in \mathscr{G'} \setminus \mathscr{G}$.
Random variables
So, the tickets also contains the values of random variables. If the random variable $X$ is $\mathscr{G}$-measurable, we get answers to all our questions about its value, such as $[$is $X$ equal to $3]$, since by $\mathscr{G}$-measurability of $X$, $\{\omega \mid X(\omega)=3\}\in\mathscr{G}$. Or, to handle the delicacies of the uncountable case, we may also ask $[$Is $X$ in the set $B]$? (Since for any particular value we think about, the probability of hearing "yes" may be $0$ and that would be boring). So, in this sense we have all information about the realization of the random variable, if our information is $\mathscr{G}$ and the RV is $\mathscr{G}$-measurable.
Caveat: the definition of measurability of random variables restricts the sets $B$ we may ask about. $[$Is $X(\omega) \in B]$ is answered if $B$ is a measurable set in the value space of the random variable (usually Borel $\sigma$-algebra is assumed with $\mathbf{R}$ without mentioning). So, in the uncountable (nondiscrete $X$) case, don't ask whether $X$ is in the Vitali set or the oracle holding the ticket shall be mad.
Reference
I did not cite any reference in the answer but I consulted
- J. Jacod and P.E. Protter. Probability essentials (2nd edition), Springer, 2004
about the definition of measurability of random variables. (And have learned these things from the same book previously, if I recall correctly).
