Relativity is a theory of frames of reference. The question says that $\Delta t$ is the time measured between two events by two synchronized clocks in frame $S$. But now, here's the catch: $\Delta t$ is actually the time between the two events as observed by $S$, as observed by $S'$.
Usually, in SR, we try and compare the flow of time and the measure of lengths, with respect to another frame. So, the $\Delta t$ in the equation above is what the person standing in $S'$ sees in his frame. Let me try and explain.
Suppose you look at it from the perspective of $S'$, with $S$ travelling relative to $S'$ with velocity $v$. Let's give them both stopwatches. Now, let two events $A$ and $B$ happen in $S$. When $A$ happens, $s$ (the observers are denoted by small letters here) starts the stopwatch. When $B$ happens, he stops the stopwatch. 562
The time between them is $\Delta t$, as we know from the question. Now, will the time between them, as seen by $s'$ be the same? No. The stopwatch will start and stop at a different time, so you get $\Delta t'$.
But now, say $s'$ is trying to see how much time elapsed according to $s$. He will see that $$\Delta t = \frac{\Delta t'}{\sqrt{1 - \beta^2}}$$
That's where we have to compare two observers instead of one, and think about how the clocks in two separate frames are synchronized.
Coming back to the question the OP asked, it does matter how clocks are 'ticking' in other frames, when you are comparing some quantity in your frame to the others. That's what the word 'relativity' means. How $S'$ behaves relative to $S$ or vice versa.
When $S'$ tries to measure what $\Delta t$ is going to be, of course he cares about how the clocks are synchronized in that frame. Because, he is trying to find out how much time has elapsed in the other frame.
$\Delta t$ in this context isn't the time that $s$ measures. Instead, it is the time that $s'$ sees has passed for $s$. And how does he 'see' how much time has passed for $s$? By finding out what time the clocks (or stopwatches) in $S$ are showing in between the two events (which includes measuring whether they are synchronized or not to get a better idea), and comparing those values with that of his own clocks.
So, that is why it matter to know what is going on in other frames. You may see a lot of bold and italics in this answer, and that is because this is a delicate concept, and you need to know what's going on exactly and not get caught up on the long sentences. 562