to make sense of this you have to look at 3 elements of the sentence.

1. Yet how often do we find that the confessional
Confessional refers to the place, or the moment, where confessions of sins are made.

2. of "a family or a neighborhood"
meaning that a family or an entire neighborhood will sometimes make confessions of sin.

3. is the "bedside of one who sees the green fields only as did the Lady of Shalott, by holding a looking-glass so that it reflects the out-of-doors."

This is a reference to the Lady of Shallot, from a poem by Tennyson, who suffered a curse, and was locked in a tower. She made tapestries only by seeing reflections of the landscape in a mirror. She could never see anything directly.

So the reference is the one to whom the confessions are made (at the besides of that person-- alluding to the idea that the confession are made besides the bed of that person). The literary reference is to the idea of confessing to one who knows by the mind's reflection, not by directly seeing.