The "double empathy problem" simply says that the fault doesn't have to be assigned strictly to one party, just as how it's unproductive to blame the intern who deleted the production database by accident. There's a systemic failing that should be understood instead.
Complicating matters is that folks with a disability are often pathologized, as the article notes, so it's far more likely that any fault will be assigned to them -- by both parties! (Internalized guilt is a hell of its own.)
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.
The TLDR is that there is an assumption dating back to the beginnings of Autism research that Autistic people have a "communication deficit".
The point of the Double Empathy Problem is that it shows that that's not true -- that when you have an Autistic Person and a Neurotypical Person interacting they each struggle to understand the other.
In other words, it's not a deficit on the part of the Autistic person -- it's a mismatch between communicators and should be recognized as a systemic failure mode not an error on the part of the Autistic. I hope that helps! :)
So what is the Double Empathy Problem? I'm still not sure what the article says after reading it twice.
Is it something like this?
The limitation of empathy, for those who understand why it is useful to practice it, and who try to practice it, is that unless you know how the other person thinks, just putting yourself in their shoes won't help. You also have to be able to model how their thoughts will go - not how your thoughts would go if you were in their situation.
So, if you want to treat people with autism empathically, first, investigate and learn how to model how they think. Once you have some model of that, you can practice empathy with them. If you try to practice empathy before you have a model, it won't be empathy - it will be misunderstanding.