Why was my answer to this question deleted?
My answer is below, for reference. I thought it was a good one - and certainly not delete-worthy:
I think that the question as written is overly broad; here, context is crucial, and the subject matter of the online community that you moderate is relevant information that you'd need to provide, in order to get a good answer.
However, let's assume that this IPS site is the online community that you moderate; but this means that we can apply the guidelines from the diamond moderator Catija:
and you may immediately refer the user to those guidelines; in other words, if that user is disputing your abilities to moderate IPS, have him or her contact the Stack Exchange staff members. There is no need for you to say anything more.
Now in general, e.g. not an IPS community, not necessarily a Stack Exchange site, I would say that the subject matter is crucial. For instance, it is natural and entirely justifiable for someone looking to ask a question about parenting, on some parenting Q&A forum, to expect that the moderators are not teenagers -- unless a teenage moderator has had relevant experience in parenting.
So, I think it's important not to conflate maturity with subject matter expertise.
If you have subject matter expertise (say, in parenting), then again, refer the user to that online community's staff members. You could briefly tell the user that you are qualified to moderate and give your reasons, but that's not really necessary.
If for some reason you are moderating an online community that doesn't have staff members that can help, then I'd recommend showing that you are qualified to moderate through your actions, e.g. with high quality answer posts, helpful comments, good conflict management, etc., rather than verbalizing it.