The question was recently posed to me, "why don't Catholics allow other Christians to receive their communion?" The truth is that I was hardly satisfied with my response. So here we are.
The Eucharist is the "source and summit" (Lumen Gentium, 11; Catechism of the Catholic Church #1324) of Catholic faith and life. It is sort of like the cherry on top, except that it is more like a solid-gold-orb on top. With the cherry on top, the main event is the ice cream. But with the Eucharist, everything else is preparation, is "on the way to" the Eucharist. It is the main event, that which is most precious.
When I approach the altar to receive communion, it is a sign of my communion with the Catholic Church, of my adherence to the teachings and truths she proclaims, of my loyalty to the Pope and the Bishops, the successors of the apostles. Receiving the host, the Body of Christ, is the epitome of my union and communion with the body of Catholic Christians with respect to all of these implications.
In addition to the perspective of communion, one ought to be in a state of grace to receive the Eucharist. This implies the practice of confession, the sacrament by which we are restored to this state of grace, by which we are picked up from our falls, in which we accept the infinite mercy and love of God. In other words, confession restores us to communion with God, of which our reception of the Eucharist is symbolic.
The Eucharist is communion. It is communion physically, literally, figuratively, and spiritually. It is a sign of my communion with God and of my communion with the Catholic Church.
For further reading, you may consult numbers 1395, 1396, 1400, and 1415 of the Catechism of the Catholic Church.
Similar issues are treated here (regarding Catholics attending other christian services) and