To actually respond, the big things are transcending. The simplest way of understanding is having a cosmology that is so large that it can fit a High 1-B cosmology or infinite dimensions into it or a character who transcends a High 1-B cosmology. That's low 1-A. 1-A is then a character of cosmology that transcends a Low 1-A cosmology or character. High 1-A does the same to 1-A, 0 does the same to High 1-A.
To transcend in this case, means to be so powerful or so large that you must go beyond our normal understanding of infinity in order to reach the same level.
I hope that's simply enough to understand.
First of all, you did not mention the layers in 1-A.
Transcending 1-A is not simply H1-A you can create an infinite hierarchy within 1-A.
Each hierarchy qualitatively transcends the one below it and so you get a 1-A+ cosmology.
Now what makes H1-A special is that even if you have a 1-A+ cosmology, transcending it means that it is not H1-A.
Because of this stuff, H1-A is similar to inaccessible cardinal, let me talk a little bit about the math of this (you don't need to know this, you just need to know that transcending 1-A+ is not H1-A).
First of all, we can create cardinals larger than Aleph-ω, I'll just give a simple example.
Aleph-ω1 = sup{Aleph-n : n ∈ R}
Well, you can go a little bit further and find much bigger cardinals.
We have 3 conditions for Inaccessible cardinal.
It must be uncountable: κ > Aleph-0;
It must be regular: which means it's not equal to the union of less than κ many sets with less size.
It must be a strong limit cardinal: whenever we have λ < κ then 2^λ < κ.
In short, no matter how many power sets you use, no matter how many Alephs you use, you still cannot reach the Inaccessible cardinal.
So no matter how much you transcend 1-A+ you still stay at 1-A+. To reach H1-A you should transcend the whole logical system of 1-A.
So neither H1-A nor 0 is like 1-A.