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

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.}

From an Anglican perspective, there is no Sacrament of reconciliation, as the Anglican Communion officially recognizes only two Sacraments, those being Baptism and Holy Communion, as provided in paragraph 2 of Article XXV of the Articles of Religion:

There are two Sacraments ordained of Christ our Lord in the Gospel, that is to say, Baptism, and the Supper of the Lord.

But despite that being the case, Anglicans do accept the value of Confession and Penance for which the scriptural basis would be John 20:21-23 and Matt. 18:17-18. These citations are, of course, from Our Lord, himself. It was related to by an old Anglican Priest, of blessed memory, that while he was commanded to practice the office of the keys, Jesus did not command all faithful to go to confession.