Well, if we're being very literal, every position someone holds is the product of reasoning. Believing a proposition is the result of being convinced. You can be convinced for good reasons or bad reasons, but as long as the brain is involved (and how could it not be?), reason is involved. In that scope, the statement is wrong because the premise is false. They were, in fact, reasoned into their belief.
Also, in a more colloquial sense, it refers to people whose beliefs were spawned by indoctrination, emotional appeals, socialization/inculturation and other things that aren't normally in the realm of 'pure reason' - and in this scope, the statement becomes a claim that you're just not going to be able to convince them of their error by using strict logical reasoning.
People, even if they don't realize it, value reason and evidence. We're thinking creatures and critical thinking represents the "best practices" of human thought.
The trick, if that's a fair word, is to get them to realize how much they value the principles of sound thought and get them to apply that to beliefs that have previously been protected from such scrutiny.
I'm not implying that it's always (or ever) easy to accomplish this, but the statement in question implies that it's simply impossible - and nothing could be farther from the truth.
Skepticism, critical thinking, logical reasoning, science - however you prefer to label the most consistently reliable tools we have for discerning reality - are ubiquitously recognized as valuable though not universally applied. We must continue to try to convince people of the value of applying these principles to every claim.
People often become convinced of things for bad reasons and are remarkably good at protecting those beliefs from critical examination - especially if those beliefs have been long-held, publicly professed and are shared by many others in that person's social network. But that doesn't mean that they are forever incapable of recognizing that they've accepted something for bad reasons.
You can, in fact, reason a person out of almost any belief - it just won't work if the person doesn't see the value of holding reasonable beliefs. The good news is that many, if not most, of the people who don't currently see that value can learn to see it and most already think they do, which is why the Socratic method is so darn useful.
It may sound insightful and it may accurately represent the frustrations of dealing with the most skeptically-challenged, fundamentalists - but it's simply not true in any useful sense and it's time for this sentiment to drift off into Bad Idea land...