So, what happens at the Breaking of Bread? Well, the Holy Spirit works there by faith. According to D.P. Williams (this is, after all, a series on the Apostolic theology of the eucharist), through faith the Holy Spirit makes the sacrament a means of grace and so Christ's 'invisible grace is imparted mediately [i.e. through the sacrament] and directly from His Person to His worshipping Saints' (D.P. Williams, 'The Lord's Table', Riches of Grace, May 1928).
That might sound a bit strange at first. We don't often think of God's grace as being given 'mediately' (i.e. through something). But bear with me a moment; what about the Gospel? Salvation comes to us through the Gospel. The Gospel comes to us through the medium of preaching. As Paul writes in Romans, 'faith comes by hearing' (Romans 10:17). So God's grace comes to us through the means of the preaching of the Gospel. That's why theologians call the preaching of the Word a means of grace.
The same thing happens with the sacraments. The Lord's Supper is not something magical; it's simply a way in which God shows us His grace and builds us up in faith, just like the preaching of the Gospel. And like preaching, the sacrament is only effective when joined with faith. It's not a magic change in the bread and wine; they stay bread and wine. It's as we receive the bread and wine in faith, God works by His Spirit to feed us with Christ. By the action of the Holy Spirit 'each believing saint partakes by faith of the virtue of [Christ's] own Body and Blood, as substantial sustenance to the spiritual life' ('Fundamental Truths Believed by the Apostolic Church', Riches of Grace, September 1928).