Most people are so different from me that maybe I can't understand why they would be hurting in a particular situation. Maybe that's what they mean by lack of empathy?
Probably that's one of the main reasons they say that.
Also, we usually don't show empathy in the same ways.
Sometimes I get so filled with empathy for someone, or for a group of people that it dominates all my thoughts. I've noticed that same tendency in some others with ASD.
I think that some of us have more, some have less empathy, that it depends on the individual. Just as some NTs have more and some have less empathy, depending on the individual.
It may be possible that when an NT has little empathy, people think well, that's just him. They see it as an individual trait. But if someone who happens to be ASD has little empathy, people assume it's because of his ASD. They assume "Oh, they are all like that." Which is mistaken.
But many of the persons I've known with AS have powerful empathy. We just express it differently. Sometimes we leave the person alone because we think that is what they need. But meanwhile that person thinks we are leaving them alone because we don't care. Sometimes we leave the person alone because when we tried to actively comfort people in the past,it went badly.
And sometimes we don't realize the other person is hurting in a particular situation because in that situation we wouldn't be hurting. Because we are different. NTs sometimes make this same kind of mistake, as well. They are not always socially perfect either! But if an NT makes a social mistake, people don't blame NTness. If an Aspie makes the same mistake, people assume it's because of ASD.