From Barnes' Notes on the Bible:
The more obvious signification is, that there is a sense in which it
may be said that the cup is blessed, and that by prayer and praise it
is set apart and rendered in some sense sacred to the purposes of
religion. it cannot mean that the cup has undergone any physical
change, or that the wine is anything but wine; but that it has been
solemnly set apart to the service of religion, and by prayer and
praise designated to be used for the purpose of commemorating the
Saviour's love. That may be said to be blessed which is set apart to a
sacred use (Genesis 2:3; Exodus 20:11); and in this sense the cup may
be said to be blessed; see Luke 9:16, "And he took the five loaves and
the two fishes, and looking up to heaven he blessed them," etc.;
compare Genesis 14:9; Genesis 27:23, Genesis 27:33, Genesis 27:41;
Genesis 28:1; Leviticus 9:22-23; 2 Samuel 6:18; 1 Kings 8:41.
Is it not the communion of the blood of Christ? - Is it not the emblem
by which the blood of Christ is exhibited, and the means by which our
union through that blood is exhibited? Is it not the means by which we
express our attachment to him as Christians; showing our union to him
and to each other; and showing that we partake in common of the
benefits of his blood? The main idea is, that by partaking of this cup
they showed that they were united to him and to each other; and that
they should regard themselves as set apart to him. We have communion
with one κοινωνία koinōnia,) that which is in "common," that which
pertains to all, that which evinces fellowship) when we partake
together; when all have an equal right, and all share alike; when the
same benefits or the same obligations are extended to all. And the
sense here is, that Christians "partake alike" in the benefits of the
blood of Christ; they share the same blessings; and they express this
together, and in common, when they partake of the communion.
The bread ... - In the communion. It shows, since we all partake of
it. that we share alike in the benefits which are imparted by means of
the body of the Redeemer. In like manner it is implied that if
Christians should partake with idolaters in the feasts offered in
honor of idols, that they would be regarded as partaking with them in
the services of idols, or as united to them, and therefore such
participation was improper.
(emphasis added by me)
The idea is that it is participating in the body and blood of Christ, not partaking of the body and blood of Christ in the sense that it's literally his body. The point of the passage is to say that when we eat the elements, we participate in their symbolism, their celebration of Christ, and their uniting the believers together. Therefore, similarly, if we participate in idol feasts, we're participating in the symbolism, celebration of the idol, and uniting that comes with it. It's a warning against participating in these things, not necessarily referencing its literal transformation into Christ's body.
EDIT: This is highlighted too by Paul in verses 25 and 26 where he says they can eat anything sold in the market (some of which had been sacrificed to idols) because “The earth is the Lord’s, and everything in it." Even though it had been sacrificed to idols, by buying it in the market they weren't participating in the idol worship it was originally consecrated for.