A $5\times 5$ grid of single-digit numbers in $\mathbb N$, with one cell empty. What number should be in the cell?
$$\begin{array}{c} x \\ y \\ p \\ q \\ r \end{array} \qquad\to\qquad x^y = pqr\quad\text{(concatenated)} $$
"[W]hat does it really mean to solve such a question?"
Well, such a question challenges us to find order amid chaos. That's what mathematics ---as the study of pattern--- is all about, so it's not a completely irrelevant mental exercise. As others mention, it could be instructive to find ways to justify any answer. (It's always possible to do that in a puzzle like this one, although not all rules are particularly "nice" ... but "nice" is subjective and not actually required.)
That said, this kind of thing is a horrible exercise for an exam. Not only is it unreasonable to expect someone to notice any pattern in a fixed time-frame, it is unreasonably-unreasonable to expect someone to notice the (so-called) "correct" pattern at all, since that's often akin to mind-reading. ("What was the author thinking?")
In any case, here's a walk-through of my thought process: After a minute or so of skimming the grid, and just before abandoning the whole thing, I happened to recognize the powers "27", "128", and "256" amid the columns of digits; then ---oh, yeah!--- "243" and "343" (which aren't always on the tip of my brain); and then ---hey!--- that could be "625" in the last column! I must be onto something! Yet ... "73" and "44" and friends aren't powers, so maybe "27" was a red herring. Then, the "aha!" moment: $2^7 = 128$. Done.
No science has anything to do with it.
I must say I agree. This question is, mathematically speaking, nonsense: for any of the possible "answers" $\{1,2,3,4,5\}$ you could easily cook up a function $f(i,j)$ that computes the correct value in all cells $(i,j)$.
Of course what is meant is for you to find a "simple" function that produces the numbers shown, where one person's "simple" might include rules like "count the number of letters in the English spelling of the number to produce the next entry" which might not be considered simple at all by somebody else.
To make the question precise, one would need to provide an expression grammar, and ask, for instance, for the expression tree of least depth that can produce all provided inputs, which can then be evaluated on the empty cell. As the question stands, it is ill-posed, a quantitatively-flavored brainteaser at best (with very little relation to mathematics), and IMO quite a poor question to ask on any kind of entrance examination intended to gauge mathematical ability.
Such problems are set up by people who have not the slightest idea of mathematics, but make a lot of money from their tests.
It is alright to create problems where some sort of "mathematical pattern recognition" is required, but such problems are infinitely more structure hiding and revealing than this idiotic example. Such problems (in disguise) might appear at the IMOs.
For the problem at hand find for each option A) – E) a formula containing only the most basic operations (as the test creator does not even know how to add fractions) that creates this value. Then using Ocam's razor choose the shortest of these.