What does it mean by saying "Asymptotically more efficient"?
First off, both algorithms "work" for all inputs.  The question is about performance.

The answers to that question are kind of crappy.  One way to say one algorithm is asymptotically more efficient than another is if there is some (problem-specific) input size such that for any larger input size the more efficient algorithm will take fewer "computational steps", usually by some abstract measure, e.g. number of comparisons.

The idea of the answers is that an asymptotically more efficient algorithm may still require more steps before that input size.  It can be the case that the asymptotically more efficient algorithm requires fewer steps for all inputs, but it doesn't need to be the case and in practice usually isn't.  So a better wording of the "correct" answer would be "$X$ will be a better choice for all inputs except possibly small inputs".

The wording still isn't that great though.  First off, many more factors go into deciding what algorithm is a "better choice", but I'll give them that the intent is clear enough in this case.  The real issue is "small" and "large".  One of my favorite papers is The Fastest and Shortest Algorithm for All Well-Defined Problems. This paper describes an algorithm which given any specification of a function and a proof that it can be computed in polynomial time will compute that function in optimal time complexity within a factor of $5$ plus an additive term.  For example, if I provided it an implementation of bubble sort as the function specification and the simple proof that it was $O(n^2)$, it would produce a sorting algorithm that was $O(n\lg n)$.  In fact, it would produce an algorithm that was $5cn\lg n + o(n\lg n)$ where $c$ was the constant factor of the asymptotically* optimal algorithm. This is amazing.  There's just one problem: the constant term &ndash; hidden in the $o(n\lg n)$ in this example &ndash; renders the algorithm almost certainly completely infeasible for virtually any real problem.  What do I mean by "completely infeasible"? I mean the heat death of the universe will happen many times over before that algorithm completes.  Nevertheless, for suitably "large" inputs it will be faster than bubble sort.  My point is it's almost certainly not physically possible to write down by any means a "suitably large" input, let alone compute on it.

At any rate, how I would word the correct answer would be: "$X$ requires fewer steps than $Y$ on sufficiently large inputs".  This is still a bit vague as there are multiple notions of "step" that could apply and an algorithm could be asymptotically more efficient by one metric and less efficient by another.  This wording avoids the value judgement of "better choice"; there are many reasons to choose asymptotically less efficient algorithms or even less efficient algorithms when constant factors/terms are specified such as cache-efficiency or implementation simplicity.

* There is a subtlety here. The asymptotically optimal algorithm may have a worse constant factor, $c$, than an asymptotically non-optimal algorithm. I think it will have the best value of $c$ for any asymptotically optimal algorithm, but it's conceivable that to squeeze out a slight gain in asymptotic efficiency, massive complexity is added that significantly increases the constant factor.