The Bible speaks of the Gospel as an aroma. It's a good aroma to those who are being saved, and a bad aroma to those who are being lost. (2 Cor. 2:16)
Why would "Good News" smell bad to some? Why would "Good News" be offensive? Or to put it another way...
What is the offense of the Cross?
The answer is Grace.
Fallen man wants to *deserve* God's favor, but the gospel says "No. There is none good, no not one. It must be a free gift of grace."
The "offense of the cross" isn't "showing people they are sinners". If you think that offends them, you aren't spending enough time around sinners. People revel in their sin all the time (though they may think their "good" outweighs their "bad").
And "the offense of the cross" isn't the "Lordship of Christ". People bow to gods and gurus and messiahs all the time.
The "offense of the cross" is Grace. The only place in Scripture where it's specifically mentioned is Gal. 5:11. Paul says:
"...if I still preach circumcision, why do I still suffer persecution? Then the offense of the cross has ceased."
See, if Paul preached Legalism, he wouldn't have been persecuted. But he preached The Cross, the substitutionary death and gift of righteousness that Grace requires.
And the hearer must repent (change his mind) about his self-righteousness.
And he must believe in the Risen Christ who said, "It is finished". He must "believe God" that his own righteousness is as filthy rags, and he must accept the free gift of God's righteousness. His sins can't be overlooked, they must be paid for by another, as a gift.
This Grace is a "stumbling stone, and a rock of offense" (Rom. 9:33; 1 Pet. 2:8), because fallen man is religious to the core with self-righteousness.
Grace blasts that self-righteousness with a laser beam of Good News from the Light of the World, and self-righteous man screams in smug hatred, and calls the Good News bad.
Until he is born again.
Then the "offense" becomes sweet music to his ears, and he sees the Good News as the most precious news he has ever heard.