TL;DR: There is no reason we should care whether our food has been sacrificed to idols.
The underlying idea here is whether one is committing idolatry, which is the belief that anything, other than God or his agents, has any supernatural power or attribute.
Believing that a rabbit's foot provides good luck is a form of idolatry, but so is believing that food that has been sacrificed to an idol is somehow different from food that hasn't.
Neither the rabbit's foot nor the food have any supernatural attributes.
To avoid something because it is reputed to have a non-existent attribute is equivalent to admitting that it does have that attribute.
For you, it shouldn't matter at all whether the food was sacrificed, any more than you should care what the phase of the Moon is today, or what constellation was in the sky at your birth.
But, if someone gives you something and explicitly or implicitly tells you it has supernatural attributes, then you should refuse it.
Not because it makes any difference to you, but because it makes a difference to the other person.
By taking it, you are implicitly endorsing and encouraging false beliefs, rather than providing the good example that every Christian should be.
But if someone says to you, “This has been offered in sacrifice,” then do not eat it, for the sake of the one who informed you, and for the sake of conscience —
I do not mean your conscience, but his. For why should my liberty be determined by someone else’s conscience?
Here "conscience" means:
as distinguishing between what is morally good and bad, prompting to do the former and shun the latter, commending the one, condemning the other
Rejecting the offered food shows the other person that you are condemning the act of sacrificing to idols.
Consider a non-religious analogy.
You are visiting Bavaria and when the waiter pours your coffee he proudly tells you that the pot he is holding was used by Hitler himself when he visited the inn in 1942.
What message does it send if you smile, say danke schön, and proceed to drink it?
What message does it send if you refuse the coffee?