The pain in your post is heart-rending. You must feel as though your whole world has stopped turning. At the moment, in these early weeks, you will be in shock.
I haven't gone through an enforced end to a relationship, but can empathise with the total shock and disbelief on the discovery of an affair. When something like this happens, it often evokes post-traumatic shock disorder - with flashbacks to the moment of discovery and terribly intrusive images.
The rational side of you will conclude that 5 months is a ridiculously short period of time to make a decision like this - even if he believes he loves the OW, this is not long enough a time to make that sort of decision.
I do not necessarily believe that he was "unhappy" and that this led to an affair. I believe your memories and those of the people close to you, far more. People in his situation often re-write history and pretend that this wouldn't have happened if they'd been happy, or say with marginally more honesty, that they hadn't realised how unhappy they were until they met the third party.
The real truth for many people is that they were neither gloriously happy nor terribly unhappy - they were just reasonably content and feeling occasionally overwhelmed by the busy pace of life and the responsibilities of career, children, money worries etc. Which accounts for probably most marriages of your duration with children the same age.
This is why he won't talk to you and give you reasons. He probably knows that you would see right through the justifications he has been deluding himself (and her) with. Most people involved in an affair need some sort of internal justification. This can be "it's a no-risk bit of escapist fun" right through to "I have obviously been unhappy and I love the other person".
Looking at your H's actions, I reckon he has got himself into a dreadfully pious mindset that he is an honest man. That means being belatedly honest with you about what's been going on and unforgivably honest to your DCs. He's probably telling himself that he is doing the right thing by everyone now by being "honest".
Well, sorry, but that is bollocks.
He deceived you for five months while he was making up his mind and is now dressing up cruelty as "honesty". He isn't so honest that he will actually talk to you about this and give you some insight, is he? The person he is actually being most dishonest with is himself.
One day, he might get that clarity that actually, your marriage was perfectly fine but like most marriages of this duration, it needed revitalising a bit. Instead of doing that though, he had an affair and left you and his children, telling himself that this was "true love".
I can't advise you how to move on in a practical sense, or even how to detach from him, but I can advise you to get some counselling on your own and to read all you can about affairs and the myths surrounding them. Do get a copy of Not Just Friends by Dr. Shirley Glass. There is a special chapter on healing alone in the wake of an affair discovery and I think this will help you.
Don't go near a counsellor who believes that affairs can never happen in good marriages, as that will compound your hurt and make you feel worse and more responsible for what has befallen you.
Above all, know this truth. You and he were equally responsible for your marriage, but he is 100% responsible for his affair. You couldn't prevent it, only he could have preserved his fidelity.