There's a lot of history of doctors/educators saying that autistic people have trouble empathizing with others, that they're just less equipped for empathy or emotional connection in general.
This turns out to be (for many autistic people) just not true; they empathize just fine but their experiences of the world and the way they express empathy and read empathy are different enough from neurotypical people that it's like a bridge that needs to be crossed, it's hard to connect the feelings in a way they can understand, or to know what's going on and when/why someone is feeling upset about something.
What double empathy does is it rephrases that problem as a mutual problem; the issue is not that autistic people are incapable of emotion (for some autistic people it's actually the opposite problem, some autistic people struggle with hyper-emotions and hyper-empathy). The problem is one of communication and understanding. Both neurotypicals and autistic people struggle to understand, read, and communicate with each other.
Double empathy is an attempt to take something that was historically a burden that autistic people had to handle entirely on their own, and to instead rephrase it as a communication problem that can be overcome by both parties learning more about each other and learning to relate to each other.
Or, TLDR:
1. People who are different from each other generally have trouble communicating with and empathizing with each other.
2. Autistic people are often different from neurotypical people.
3. Therefore, autistic empathy is often a problem of communication and understanding, not a one-sided moral or emotional defect within the autistic individual.