What is the Catholic/Anglican understanding of how the Sacrament of Reconciliation began? How did the Sacrament of Reconciliation begin?

I have tried to find this answer but can not trust Wikipedia and do not need any more answers for when it came to be.
Your title implies that you are looking for an answer from a "Catholic-Anglican" perspective, but perhaps you will also accept an answer from the Eastern Orthodox perspective, which shares a common early Church tradition.

In the Orthodox understanding, the Sacrament of Reconciliation (or Repentance) was instituted by Christ Himself:


  The Lord instituted the Mystery of Repentance after His Resurrection,
  when, having appeared to His disciples who, except for Thomas, were
  gathered together, He solemnly said to them: Peace be unto you....
  And when He had said this, He breathed on them, and saith unto them:
  Receive ye the Holy Spirit. Whosesoever sins ye remit, they are
  remitted unto them; and whosesoever sins ye retain, they are retained
  (John 20: 21– 23). Moreover, even before this, Christ the Saviour
  twice uttered a promise about this Mystery. The first time He said to
  the Apostle Peter, when Peter, on behalf of all the Apostles, had
  confessed Him to be the Son of God: I will give unto thee the keys of
  the Kingdom of Heaven: and whatsoever thou shalt bind on earth shall
  be bound in heaven; and whatsoever thou shalt loose on earth shall be
  loosed in heaven (Matt. 16: 19). The second time He testified to all
  the Apostles: If he neglect to hear the Church, let him be unto thee
  as a heathen man and a publican. Verily I say unto you: whatsoever ye
  shall bind on earth shall be bound in heaven; and whatsoever ye shall
  loose on earth shall be loosed in heaven (Matt. 18: 17–18).1


This accords with how the "Sacrament of Penance and Reconciliation" is explained within the Vatican's Catechism of the Catholic Church 2.2.2.4.VI, which cites these same passages.

The commentary on Canon XI of the First Ecumenical Council in the Nicene and Post-Nicene Fathers series provides an overview of the how epithemia (penance) was viewed and administered in the early Church.  This can be summarized:2


  In the ancient Church there existed a rite of public repentance for
  the “fallen,” and in particular for those who had not held firm in the
  faith during the persecutions. According to this rite, the penitents
  were divided into four classes: (a) the “weepers,” who did not have
  the right to be present at the public Divine services and, stretching
  out their hands off the church porch, with weeping would beg those who
  entered the church to pray for them; (b) the “hearers,” for whom it
  was permitted to be in the narthex of the church all the way to the
  end of the liturgy of the Catechumens; (c) the “prostrators,” who
  entered the church itself but also did not participate in the Liturgy
  of the Faithful (after the Liturgy, on bended knees, they were
  vouchsafed the pastoral blessing); and (d) the class of those who
  “stood together” with the faithful for the whole Liturgy, but could
  not receive Communion of the Holy Mysteries.


The First Ecumenical Council was recognized by all five of the ancient Sees in antiquity, including Rome.



1  Michael Pomazanski, Orthodox Dogmatic Theology (3rd ed.), p.293
2 Ibid.