Yes gladly, although I am sad for you that your H is in such denial and doesn't seem to be willing to help you with this. It shouldn't be you struggling to understand this, it should be him. Are you being too willing and appeasing about this, because it sounds like it to me?
The primary addiction is to the feelings this correspondence has generated. I've got no doubt that they will have reminisced about their romance, the sex, their feelings at the time and of course, their youth, the people they knew and the era they lived in. There will also have been references to how gorgeous/beautiful they thought the other was.
This would have progressed to lots of positive strokes and mirroring. So your H was probably getting lots of positive affirmations about himself. They might have discussed their current lives and views about issues; their hobbies and interests. What happens then is a phenomenon best described by the phrase "me too!" in that they will have pretended to feel the same way and like the same things.
Unfortunately, this can extend to their descriptions of how happy and content they are in their current lives. If this woman had confided that she was unhappy and married to a complete arse, your H might have known at an instinctive level that he couldn't say the same in reverse, but would have told himself that it would have been impolite and gloating to express happiness in his marriage, so he might have said "me too!" or spoke about you and his marriage in bland, ineffusive terms - phrases like "we're fine", "we're happy enough" or "we rub along together reasonably well".
The "not wanting to appear gloating" bit is a complete delusion however and the real reason that people don't effervesce about their happy marriages is because they want the other person to believe that there is a gap that s/he can fill.
By now, he would have become addicted to all this positive stroking and mirroring and will also have developed a secondary addiction the the thrill of actually logging in and finding a message, or hearing his phone vibrate or ping to signal a communication from her.
Lots of people are addicted to the technology and the buzz of receiving comments or messages. Think of Mumsnet, or think of the people you know who are constantly playing with their i-phones. Giving up on any of these things means losing out on the buzz or thrill, which can be as enjoyable as reading the communications at times.
I expect he has become addicted to the feelings this has evoked and addicted to the communication itself. He has enjoyed the feeling that someone (else) out there adores and respects him.
But it's all a mirage. If she were married to him, she would get irritated by the same things as you do. If he were married to her, he would get irritated by her foibles too. It's a fantasy relationship, but try telling the participants that, when they are in the grip of something like it. It really is absurd and ridiculous when you analyse it properly.