Naturally, most don't reject the idea of reconciliation itself; who would, save for those who proudly refuse to grant that they need to be reconciled with God and/or neighbor? What so many abhor is the confession bit—specifically, frank confession of their sins to another human being in private. Most people don't even like doing that with those they love, much less to a priest.
I don't think most such people refuse on principle to admit their sins. If they have insight and honesty enough to know how they've been wrong, they'll usually admit it sooner or later. Neither do most people fear that the seal of the confessional will be broken and, with it, their confidence in the Church. The problem, at least as they often put it, is that they cannot accept the idea that another sinful, fallible human being—in this case, a priest—has the right and duty to mediate God's mercy to them.
I have always found that attitude so silly as to constitute a flimsy rationalization. When spouses, friends, or co-workers forgive one another after a candid airing of grievances, they are mediating God's mercy to each other. Why, then, is it so hard to accept that from a man ordained of God?
Part of it, in my experience, is that they hate confessing to a stranger or near-stranger. Sure, we can eat humble pie with those we know and love, or at least admit the need to do so even when we lack the courage. But somehow, doing it in front of somebody who seems to have nothing to gain or lose from it seems like a pointless, institutional requirement that one can do without, thank you very much. The problem here is what psychologist of religon Gordon Allport called that of "heteronomy." We don't take well to having rules set for us by people who owe us neither money nor love.
Yet if one believes that the Catholic Church is who she says she is, all such sentiments must be overcome. The sacrament is always there and eventually needful, supplying grace in the form of a personal encounter with a man acting in persona Christi and thus with Christ Himself. But many ill-catechized Catholics don't like the idea that the whole Christ is the risen Christ, the Head, plus his Mystical Body, the members of the Church. Let's just cut out the sinful middlemen and go directly to the Head. That, I believe, was the primary spiritual impetus for Protestantism nearly half a millennium ago. This is a problem of pride and catechesis, with the former needing to be discredited by the latter.
But even for many good Catholics, the sacrament seems like a sore spot. They're either too scrupulous about the state of their souls or not scrupulous enough. The whole issue is a nodal point for neuroses, even aside from moral theology. Such people are needlessly shortchanging themselves. And that is as good a proof as any that the Devil is very active in coming up with ways to stop people from frequenting this sacrament.
For sound advice on how to get the most out of confession, see the Pontificator's quote from John Henry Newman and Fr. Martin Fox's coordinate reflections at Bonfire of the Vanities. There's a lot of good literature out there on this topic, but those blog posts should be enough for the troubled to start with.