The Lord’s Supper is a symbol and sacrament of the believer’s union and participation in Christ and in His death. In 1 Corinthians 10, Paul says, “The bread which we break, is it not the communion of the body of Christ?” The word “communion” there indicates a communal participation, that we as a fellowship, as a body, receive from Christ the blessings and benefits of His life, death and resurrection. This is what it means to eat His flesh and drink His blood- that by faith, we are strengthened and blessed by the life of Christ, worked in us spiritually.
When we understand this concept, we can also understand why we fence the table. “Fencing the table” refers to the practice of the Reformed (and other) churches of inviting only those who are members of the Christian church to join us in the celebration of the Lord’s Supper. For the church to knowingly distribute the elements to those who are not part of His body is to break the message of the Supper, to destroy the symbol of feeding on Christ, since we are actively involving people who cannot feed on Christ, since they do not have faith in Him. Therefore Paul warns us (1 Corinthians 11:28-29) that only those who have examined themselves and are capable of discerning the body of Christ are participating in a worthy manner. “Discerning the body” means recognizing the nature of the body of Christ, the church, and the fellowship the church has with Christ.
We are not to judge people’s hearts. We cannot discern if someone has true faith. But one result of true faith ultimately will mean joining with God’s people in the church and coming under the government of the church. Since the church cannot judge the heart, it can only deal with what it sees, and therefore the Reformed church has long used church membership as evidence that someone is in fellowship with Christ, as far as we can see.
This also agrees with the nature of the sign, since it is a communal participation. Since it is the body of Christ which is communing with Him, and the church is the body of Christ, those that are members of that body are those that rightly commune. Since we cannot know who is part of the church inwardly (since we cannot judge the heart), we must judge who is part of the church outwardly, by membership.
This is why we fence the table. We do so not to condemn or judge anyone, but simply to guard the purity of the symbol that the Lord has given us, as He Himself taught us to do. God alone judges the heart, but the church is called to be faithful in the outward administration of the signs and seals that God has given us.