What you've got to think is that $X_n$ is the value of some random variable at time $n$. So, the "remote future" is what happens "after a very large time", that is, for $n$ very large. For convenience, I will assume my time unit as years, so $X_0$ is the value now, $X_1$ is the value after $1$ year, $X_2$ is the value after $2$ years (of some quantity, say price of a teddy bear at my favourite toy shop today is $X_0$, after $1$ year is $X_1$ , after two years is $X_2$ and so on).
Let us take examples to clarify this.
For example, consider $\mathcal F_{23}'$. This consists of events that can be determined by purely knowing only the values of the process after $23$ years from now. So , even if you have absolutely no clue about this event for at least $23$ years from the start, you will be able to figure out whether this event occurs or not by observing the values of $X_n$ starting from $X_{23}$.
Another way to say the same thing , but a way more convenient for usage : if I change the values $X_1,...,X_{22}$ to any value I like, the probability of any event in $\mathcal F_{23}'$ will be unaffected. That's because these events depend entirely upon the variables $X_{23},X_{24},\ldots$, so you don't need to observe anything for the first $23$ years if you want to study such an event.
For example, the event : will $X_{24}$ be greater than $76$, is in $F_{23}'$, because it depends on something that can't be observed within $23$ years of the start. You have no clue about $X_{24}$ until $24$ years.
So is the event : will $X_{28}$ be bigger than $X_{69}$?
But the event $X_{11} < X_{27}$ won't be in $F_{23}'$ since we have $X_{11}$ here, which depends on something happening before $23$ years are up. After $11$ years, I will know the value of $X_{11}$ : so for example, if it is small, I will know that it is very likely that $X_{27}$ is going to be larger than it, so you have a clue about the event before $23$ years. Alternately, if I change $X_{11}$ to something very small, this probability increases, so this event can't be in $\mathcal F_{23}'$.
Similarly, the event $X_7 + X_8 > 21 X_{29}$ depend on values before $23$ years, so they won't be in $\mathcal F_{23}'$. If I change $X_7,X_8$ this probability is going to change, right?
Now, while it is slightly difficult to believe initially, it is definitely true that there are events, that don't depend on events happening before $23$ years, before $24$ years, before $500$ years, and so on.
These events are events that lie in $\mathbb F_n$ for every $n \geq 1$. In particular, they don't depend on anything any number of years from now, but instead depend on every possible distant future.
And with the other interpretation, what you get is this : If I pick any $X_{n_1},...,X_{n_k}$ for any $n_1,...,n_k \geq 1$ and change their value, the event does not change.
For example, let us take an event from what you have, defined by $\{X_n \in B_n i.o.\}$.
Now, changing finitely many values of the $X$s is not going to affect this event, because that will reduce/increase the number of $X_n$ that are in $B_n$ by only a finite number, so the total number of $X_n$ still in $B_n$ will either remain infinite, or remain finite as per the initial configuration of the $X_i$.
For example, if I change say $X_1,X_2,...,X_{2376}$ to some values which are not in $B_1,B_2,...,B_{2376}$ respectively, then the event doesn't change, because if it were true, then $X_n \in B_n$ is happening for infinitely many $n$, and removing $2376$ of these is not going to affect the infiniteness.
Such events are part of what we call the tail sigma algebra. You have to think of the tail like the tail of a sequence : does the convergence of a sequence depend upon its first $50,000$ terms, for example? Its first $10^{10^{10}}$ terms? No. Properties which behave "like this" would be those in the tail sigma algebra.
Let us take the next example, $\lim S_n$ exists. Indeed, existence has nothing to do with the first finitely many terms, right? If $S_n$ existed, and I made all of $X_1,...,X_{2354}$ very large, then the sum will still exist. If $S_n$ did not exist, and I made all of $S_{45},S_{24},S_{456456}$ small, it is not going to help : the sum still won't exist.
That makes it a tail event.
Now, why is the event $\limsup S_n > 0$ not a tail event? Well, because unfortunately , when you take a sum of terms, every non-zero term contributes, and unfortunately the largest term contributes the most. So, for example, imagine that I had $\limsup S_n <0$. By making $X_1$ very very large, each of $S_1,S_2,...$ includes $X_1$ in its sum, so it is possible that I can lift the limit superior of $S_n$ over zero by taking $X_1$ sufficiently large.
Look at this from the series point of view. Suppose you have an infinite series which sums to $-1$ which is less than $0$. By increasing the first term by $2375$ , I will make the sum $2374$, which is greater than $0$.
So the sum being negative is not a tail event, because increasing /decreasing a few finitely many terms (above, I changed only the first term $X_1$) can potentially make it negative/positive!
Naturally the most important result regarding the tail sigma algebra has to be Kolmogorov's theorem : if $X_i$ are independent events, then any event in the sigma algebra has probability zero or one.
So what makes the tail sigma algebra so powerful is the assertion that any event belonging in it has an easy to find probability : if you can show it has non-zero probability, then it has probability one.
For example, if $X_n$ are independent , then the tail sigma events we discussed above would have probability zero or one (which one? That depends upon what the $X_n$ are, of course).
Note : I must point out that in my earlier version of this answer, I had replace the words "can be determined by observing $X_{23},X_{24},...$ alone" with the actually incorrect phrase "does not depend upon the values of $X_n$ before $23$ years". The words "depend" and "determine" are what create the trouble here.
So, what is the precise formulation? Well, let's first take the most obvious events that belong in $\mathcal F_{23}$ (for example) : For a Borel real-valued function $f$ on $m$ variables, a list of values $i_1,i_2,...i_m$ all greater than (or equal to) $23$, and a Borel set $B$, the set given by $\{(\omega_1,\ldots,\omega_n) : f(X_{i_j}(\omega)) \in B\}$ explicitly "depends" upon the values $X_{i_1},...,X_{i_m}$ for each index bigger than $23$, so these events are what we naturally consider as belonging in $\mathcal F_{23}$.
It turns out that this set of sets detailed above, "generate" $\mathcal F_{23}$ : that is, after taking all the sets described above for various $f$, indices $i_j$ and $B$, and then taking the complements and infinite unions among each of these sets, we will arrive at $\mathcal F_{23}$.
To recap, what this means is that an event in $\mathcal F_{23}$ can be explicitly created from a bunch of sets, each of which occur or not, based alone on the knowledge of some values at least more than $23$ years from now.
This, however, DOES NOT mean, that we cannot even hazard a guess about the probability of an event in $\mathcal F_{23}$ , knowing the values of $X_{1},X_{2},\ldots,X_{22}$. For example, if you take the most rudimentary process, the constant process where $X_1 = X_2 =X_3 = \ldots$, then every event in $\mathcal F_{23}$ can be determined from just the knowledge of $X_1$ alone, rather obviously! So saying that events in $\mathcal F_{23}$ don't depend upon the values $X_1,X_2,\ldots,X_{22}$ is wrong : BUT saying that such events can be determined purely by knowing $X_{23},X_{24},\ldots$ is correct (and is what the definition says).
I thank @900edges in the comments for spotting and help correct this very important error.
