Where is the concept of “local church” from?
First of the whole concept of local churches is something that all Christian denominations can refer to according to their proper traditions.
I will be using Catholic sources to demonstrate that the very nature of local churches was something well founded in Apostolic times. The history of this notion has changed only in the sense that various church communities such as in protestantism have different views on what the modern version of what local churches entail!
Regardless of the fact that I am employing Catholic sources, the biblical references to such a concept of local churches can not be ignored.
The term church (Anglo-Saxon, cirice, circe; Modern German, Kirche; Swedish, Kyrka) is the name employed in the Teutonic languages to render the Greek ekklesia (ecclesia), the term by which the New Testament writers denote the society founded by Our Lord Jesus Christ. The derivation of the word has been much debated. It is now agreed that it is derived from the Greek kyriakon (cyriacon), i.e. the Lord's house, a term which from the third century was used, as well as ekklesia, to signify a Christian place of worship. This, though the less usual expression, had apparently obtained currency among the Teutonic races.
The term ecclesia
In order to understand the precise force of this word, something must first be said as to its employment by the Septuagint translators of the Old Testament. Although in one or two places (Psalm 25:5; Judith 6:21; etc.) the word is used without religious signification, merely in the sense of "an assembly", this is not usually the case. Ordinarily it is employed as the Greek equivalent of the Hebrew qahal, i.e., the entire community of the children of Israel viewed in their religious aspect. Two Hebrew words are employed in the Old Testament to signify the congregation of Israel, viz. qahal 'êdah. In the Septuagint these are rendered, respectively, ekklesia and synagoge. Thus in Proverbs 5:14, where the words occur together, "in the midst of the church and the congregation", the Greek rendering is en meso ekklesias kai synagoges. The distinction is indeed not rigidly observed — thus in Exodus, Leviticus and Numbers, both words are regularly represented by synagoge — but it is adhered to in the great majority of cases, and may be regarded as an established rule. In the writings of the New Testament the words are sharply distinguished. With them ecclesia denotes the Church of Christ; synagoga, the Jews still adhering to the worship of the Old Covenant. Occasionally, it is true, ecclesia is employed in its general significance of "assembly" (Acts 19:32; 1 Corinthians 14:19); and synagoga occurs once in reference to a gathering of Christians, though apparently of a non-religious character (James 2:2) But ecclesia is never used by the Apostles to denote the Jewish Church. The word as a technical expression had been transferred to the community of Christian believers. - The Church (Catholic Encyclopedia)
The most natural concept of local churches can be visualized more easily in St. John’s Apocalypse where he writes to the seven churches of Asia.
In the Apocalypse the "angels" to whom the letters to the seven Churches are addressed are almost certainly the bishops of the respective communities. Some commentators, indeed, have held them to be personifications of the communities themselves. But this explanation can hardly stand. St. John, throughout, addresses the angel as being responsible for the community precisely as he would address its ruler. Moreover, in the symbolism of chapter 1, the two are represented under different figures: the angels are the stars in the right hand of the Son of Man; the seven candlesticks are the image which figures the communities. The very term angel, it should be noticed, is practically synonymous with apostle, and thus is aptly chosen to designate the episcopal office. Again the messages to Archippus (Colossians 4:17; Philemon 2) imply that he held a position of special dignity, superior to that of the other presbyters. The mention of him in a letter entirely concerned with a private matter, as is that to Philemon, is hardly explicable unless he were the official head of the Colossian Church. We have therefore four important indications of the existence of an office in the local Churches, held by a single person, and carrying with it Apostolical authority. Nor can any difficulty be occasioned by the fact that as yet no special title distinguishes these successors of the Apostles from the ordinary presbyters. It is in the nature of things that the office should exist before a title is assigned to it. The name of apostle, we have seen, was not confined to the Twelve. St. Peter (1 Peter 5:1) and St. John (2 and 3 John 1:1) both speak of themselves as presbyters". St. Paul speaks of the Apostolate as a diakonia. A parallel case in later ecclesiastical history is afforded by the word pope. This title was not appropriated to the exclusive use of the Holy See till the eleventh century. Yet no one maintains that the supreme pontificate of the Roman bishop was not recognized till then. It should cause no surprise that a precise terminology, distinguishing bishops, in the full sense, from the presbyter-bishops, is not found in the New Testament. - Catholic Encyclopedia
We also see St. Paul writing to various local churches during his lifetime:
In the Pastoral Epistles the new situation appears even more clearly. The purpose of these writings was to instruct Timothy and Titus regarding the manner in which they were to organize the local Churches. The total absence of all reference to the spiritual gifts can scarcely be otherwise explained than by supposing that they no longer existed in the communities, or that they were at most exceptional phenomena. Instead, we find the Churches governed by a hierarchical organization of bishops, sometimes also termed presbyters, and deacons. That the terms bishop and presbyter are synonymous is evident from Titus 1:5-7: "I left thee in Crete, that thou shouldest . . . ordain priests in every city . . . For a bishop must be without crime." These presbyters form a corporate body (1 Timothy 4:14), and they are entrusted with the twofold charge of governing the Church (1 Timothy 3:5) and of teaching (1 Timothy 3:2; Titus 1:9). The selection of those who are to fill this post does not depend on the possession of supernatural gifts. It is required that they should not be unproved neophytes, that they should be under no charge, should have displayed moral fitness for the work, and should be capable of teaching. (1 Timothy 3:2-7; Titus 1:5-9) The appointment to this office was by a solemn laying on of hands (1 Timothy 5:22). Some words addressed by St. Paul to Timothy, in reference to the ceremony as it had taken place in Timothy's case, throw light upon its nature. "I admonish thee", he writes, "that thou stir up the grace (charisma) of God, which is in thee by the laying on of my hands" (2 Timothy 1:6). The rite is here declared to be the means by which a charismatic gift is conferred; and, further, the gift in question, like the baptismal character, is permanent in its effects. The recipient needs but to "waken into life" [anazopyrein] the grace he thus possesses in order to avail himself of it. It is an abiding endowment. There can be no reason for asserting that the imposition of hands, by which Timothy was instructed to appoint the presbyters to their office, was a rite of a different character, a mere formality without practical import.
With the evidence before us, certain other notices in the New Testament writings, pointing to the existence of this local ministry, may be considered. There is mention of presbyters at Jerusalem at a date apparently immediately subsequent to the dispersion of the Apostles (Acts 11:30; cf. 15:2; 16:4; 21:18). Again, we are told that Paul and Barnabas, as they retraced their steps on their first missionary journey, appointed presbyters in every Church (Acts 14:22). So too the injunction to the Thessalonians (1 Thessalonians 5:12) to have regard to those who are over them in the Lord (proistamenoi; cf. Romans 12:6) would seem to imply that there also St. Paul had invested certain members of the community with a pastoral charge. Still more explicit is the evidence contained in the account of St. Paul's interview with the Ephesian elders (Acts 20:17-23). It is told that, sending from Miletus to Ephesus, he summoned "the presbyters of the Church", and in the course of his charge addressed them as follows: "Take heed to yourselves and to the whole flock, wherein the Holy Ghost has placed you bishops to tend [poimainein] the Church of God" (20:28). St. Peter employs similar language: "The presbyters that are among you, I beseech, who am myself also a presbyter . . . tend [poimainein] the flock of God which is among you." These expressions leave no doubt as to the office designated by St. Paul, when in Ephesians 4:11, he enumerates the gifts of the Ascended Lord as follows: "He gave some apostles, and some prophets, and other some evangelists, and other some pastors and doctors [tous de poimenas kai didaskalous]. The Epistle of St. James provides us with yet another reference to this office, where the sick man is bidden send for the presbyters of the Church, that he may receive at their hands the rite of unction (James 5:14). - Catholic Encyclopedia
But why have local churches?
It is obvious that at the local level, the Church in one area can be very different than the one in the next in the next city. This would be more pronounced in the Apostolic Times through to the end of the Middle Ages, when dialects of various languages would be different than those of the regional cities and big towns. This says nothing of local traditions or local Christian festivals that pertain uniquely to a local church, whether of a particular country or region. We may be Christians, but at the same time we are all unique and we celebrate things somewhat differently than those in the next place over.
Some local churches celebrate feasts that are particular to their location, which are in fact noted in the Scriptures. For example in Malta they celebrate the Shipwreck of St. Paul on February 10th.
It matters not what denomination we are. The way we celebrate our spirituality is formed from the local church we live in.
How individual denominations and/or communities incorporate the term of local churches will vary according to their concept of traditional biblical thought to modern interpretation of this subject. Some preachers could possibly believe that the local church holds all authority, whereas most major denominations hold that the local church is united to a more centralized from of Church governance, such as Catholicism, Orthodoxy and Anglicanism.