I think, ultimately, it requires empathy, an innate sense based on past familiarity with that person, whether they are genuine or not... I ask myself: How many times has my bullshit meter gone off with this person? If hardly ever, then I find it much easier to forgive. An effective apology requires that the recipient senses sincerity and contrition, commensurate with the nature, or degree, of the perceived offence.
I think it is important not to turn the offender's feelings of humility into those of humiliation. The only result can be that they may feel disinclined to express such humility to others that they may offend in their future... it sets a bad precedent for them. Again, I find it is the less cognitively able and empathetic that tend to lack this insight , relying, instead, on their emotional response alone to be judge and jury. I might also add that it is also age and maturity that is a factor; I do expect very young people to lack this, regardless of mental abilities..