The phrase 'the Body of Christ' can refer to three things - 1. the body of Jesus of Nazareth before he was crucified; 2. the community of believers; 3. the bread consecrated during the Eucharist.
In practice we can ignore 1 as it never figures in debates about communion. What is significant is the way in which the other two senses have been understood in Christian history.
Let's call those two senses of 'the body of Christ' 'the church' and 'the host'.
In Christian understanding, one form of the body was 'real' or 'true'. In other words it was something that could be touched and handled, and was therefore worthy of reverence and immense - total! - respect. This was called the 'corpus verum'.
The other form of the body was only perceptible to the eyes of faith, it could only be received and understood mystically, in the context of prayer and worship. This was called the 'corpus mysticum'.
For the first thousand years or so of Christianity, the 'corpus verum', the body that could be touched and handled with reverence, referred to the church, ie the community of the baptised. So, your neighbour in the community was worthy of reverence and respect. Harming your neighbour, eg murder, wasn't just immoral, it was blasphemy. Correlative with that, the 'corpus mysticum' - that which could only be perceived with the eyes of faith - was the host, that which was consumed in the context of Eucharistic worship.
In the course of the twelfth century, in the Western church, these meanings were reversed, with awful consequences.
To begin with the more trivial, the 'corpus verum' began to be used to refer to the bread used in the Eucharist. Instead of this bread being something that could only be seen as holy by the faithful (and which didn't have a particular tangibility as the body) the host became _itself_ the object of worship. This can be seen through the institution of the feast of Corpus Christi in the mid-thirteenth century, and the associated development of eucharistic devotions, eg exposition, seen through the use of the monstrance - the Body of Christ is being _demonstrated_ in this rite.
I happen to see this as a profound distortion of Christianity, but I needn't detain you with that, for the really malefic consequences of this shift came with the other side, ie that instead of all the baptised being the 'corpus verum', now the baptised were the 'corpus mysticum' - which had the consequence that church membership was no longer something public, it was something private, and only accessible to those with the eyes of faith. Of course, those 'eyes of faith' became identified with the institution, so, whereas harming a baptised believer would once have been utterly unthinkable theologically, with this shift in understanding you end up with the Inquisition - abuse of the body to try and establish the state of the soul. You also lay the seeds for the Reformation, and the whole gamut of western history that sees faith as something 'private' and personal, rather than public and visible.
It would be no exaggeration to say that everything that has gone wrong with Western Christianity since the 1200s can be traced to this shift.