I have a paper due and it is about critiquing one of Nietzsche's statements. I have to provide an example for my reason and defense for his.
First of all, having a paper due means researching sources and reading a lot in the first place. It has to be your original effort, otherwise, you can fail the module. Considering the question, commentaries and papers that involve aphorism 196 are to be found. In the light of this, I deliberately refer to a single source here, a source which can be used but should be weighed against other, preferably more recent publications. We cannot and should not make this work for you.
First step: Understanding the aphorism
You cannot critique or defend something well if you do not fully understand the content. Thus, you have to identify the point he is trying to make.
Monika Langer in her book Nietzsche's Gay Science: Dancing Coherence (Springer:2010) has the following to say (p. 158):
Unfree spirits also tend to follow anybody running in front of them – even someone actually ﬂeeing them. Nietzsche declares unfree spirits are thoroughly “herd”. We saw earlier such adherents ironically pride themselves on being more virtuous than the one they are following. In unmasking various facets of such alleged virtue, Nietzsche tacitly urges us to question ourselves and our age. However, he observes we hear solely questions we can answer. Rushing through life renders us incapable of ﬁnding answers requiring reﬂection.
Nietzsche advises caution in interacting with unfree spirits. They are deaf to numerous questions and happily communicate secrets they claim to conceal.
Thus, her commentary suggests that "we" is alluding to the common man, the unfree spirit incapable of true reflection. They will prise themselves for the mainstream (i.e. "herd") knowledge they are able to reproduce and simply ignore questions that are out of the box and truly questioning the status quo. Thus, one can say that "we" - common humans - "do not hear" - i.e. deliberately ignore - "questions to which we are [in]capable of finding an answer".
Second step: Critique
Critique always includes consideration of arguments in favour of the position or point if it is to be well-graded. Thus, you should try your best to think of - and find sources for - straightforward arguments for and against such a view. I will give some hints which obviously need work and references put into them - since this is what you get your grade for.
First of all, the aphorism cannot be literally true since otherwise, we would not be able to develop new knowledge, either individually or culturally. Nietzsche was aware of that, but critical towards what he conceived to be a sheepish and repetitive culture without any true progress.
What are the possible merits of the aphorism? Some have already been discussed in comments or answers:
Confirmation bias is a psychological fact, people tend to ignore input that does not sit well with their already established knowledge and worldviews.
The inability to understand something that cannot be framed in one's conceptual framework is an epistemological argument. There are plenty of authors defending such a view.
One might want to add the social dimension Nietzsche is often suggesting as well: People thinking out of the box and genuinely questioning the norms and common knowledge tend to be pariahs. Nobody really likes them either because they are so much beyond our own abilities and understanding (genius, Übermensch - envy) or because they make us feel bad for adhering to what they can show us to be wrong (projected self-hatred).
Thus, indeed only few people want to be confronted with these kinds of questions - and so they are commonly ignored by "not hearing" them.
A word on Nietzsche and the kind of philosophy he offers
Given that all this analysis is correct, why didn't he just write that and gave proper arguments? Mainly because he (just as e.g. Schopenhauer) deemed this to be exactly the style so typical for "the herd" and thus beneath his dignity. He thought the argumentative, structured philosophy of Kant to be the epitome of abhorrence (see only two aphorisms earlier). He wrote against such technical, yet herdish philosophy. Accordingly, Nietzsche writes in aphorisms, which are a literary form, a form of the art of writing. Asking him for proofs is misplaced.
The complete aphorism 196 from Nietzsche's The Gay Science is what the OP quoted:
We hear only the questions to which we are capable of finding an answer.
The OP wants to know what Nietzsche's justification for this aphorism might be.
A literal reading of this aphorism suggests that it is not true. One can find questions in a textbook on a subject that we have not studied and actually hear those questions without having any idea of how to answer them. So we can literally hear questions we cannot answer.
What we are not able to do is understand those questions, or at least understand them well enough to suggest how we might try to provide an answer if we wanted one. What Nietzsche may have intended by the word "hear" is "understand". Given that, it still does not seem true. We may be able to understand the question whether an unsolved mathematical conjecture is true or not without being able to find a proof of, or counter-model for, the conjecture.
Perhaps Nietzsche means completely understand a question defining complete understanding as actually being able to find an answer if we wanted to? If that is the case we might be able to rephrase the aphorism as
We completely understand only the questions to which we are capable of finding an answer.
However, that does not sound very profound. It just repeats our definition of "completely understand".
We could then go back to the original aphorism and suggest that Nietzsche's justification for stating the aphorism at all was to get us to think about the relationship between questions, answers and our hearing or understanding them without providing an answer.
Defense for his statement:
We rely on incoming sense data to formulate a model of the world and act according to that model. But our formulation of the world is heavily constrained by our five senses.
We only see through a tiny slit of reality because of this limitation, therefore our model of the world is simply a rough approximation, which has been developed through the process of evolution for us to survive.
Since our perception of reality exists in this rough approximation of the world that we have conjured, our rationality is also confined within this box. We are incapable of conceiving questions beyond what we are "capable of finding answers to", because the act of asking a question implies one "knows what he does not know".
One cannot ask a question which lies outside the bounds of his conception of reality, because he does not know what he does not know.
What then is truth? A movable host of metaphors, metonymies, and anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished
For me implies we can only answer a question by making sense of its metaphorical structure, and metaphors are rich verbal expressions that leave most things implicit
A sentence metaphor typically likens many things or kinds to many other things or kinds at a single verbal stroke. Benjamin’s terse little aphorism manages to liken works to death masks, conceptions to living human beings, the changes a conception undergoes before being incorporated into a finished work to life, the stabilization and stultification it allegedly undergoes after such incorporation to death—and so on.
Because of Nietzche's anti-essentialist "perspectivism", the truth is a matter of subjective attitudes, and given the ubermensch ideal has no human limits, we may find that the metaphorical structure of question, at least about value, is limitless and un-grounded. Our attempt to understand it will be a reflection of our own triviality or concern
if you gaze long enough into an abyss, the abyss will gaze back into you