Anybody have a clue what he means by all this?

I think that he is essentially saying the he believes in the idea of non overlapping magisteria. It should be fairly obvious that this is a case of compartmentalization. Here is a related post religions claim to be nondisprovable

Here are some other quotes from the interview not in the OP:

Religion is very different from science. The main part of religion is not about the way that we model the real world. I am purposely using the word “model”. Religion is an experience mainly an emotional and aesthetic one. It is not about whether the earth is 5,765 years old. When you play the piano, when you climb a mountain, does this contradict your scientific endeavours? Obviously not. The two things are almost though not quite orthogonal. Hiking, skiing, dancing, bringing up your children you do all kinds of things that are almost orthogonal to your scientific endeavour. That is the case with religion also. It doesn’t contradict; it is orthogonal. Belief is an important part of religion, certainly; but in science we have certain ways of thinking about the world, and in religion we have different ways of thinking about the world. Those two things coexist side by side without conflict.

This is an example of what is behind the figure of God call it a model, a way of thinking, a way of living. It is similar, broadly speaking, to the earth being round. God is a way of thinking of our lives; translated into practical terms, it tells us how to live as human beings.

Truth is in our minds. If we are sufficiently broad-minded, then we can simultaneously entertain different ideas of truth, different models, different views of the world.

Do you think this is a valid way of looking at the world and/or religion? If not, how confident are you in your assertion? If you are very confident, on what basis do you think you have greatly out-thought Robert Aumann?

No. I won’t go into details, but see crisis of faith which also talks about Aumann. I also don’t think it’s a matter of out-thinking, per say, I think that it is more of a matter of clearer thinking. I am not a religious authority or been brought up with a religion, so I can see it from the outside. I think that a big reason for Aumann’s belief is the costs involved if he were to start to disbelieve. See the below quotes:

The observance of the Sabbath is extremely beautiful, and is impossible without being religious. It is not even a question of improving society it is about improving one’s own quality of life. For example, let’s say I’m taking a trip a couple of hours after the Sabbath. Any other person would spend the day packing, going to the office, making final arrangements, final phone calls, this and that. For me it’s out of the question. I do it on Friday. The Sabbath is there. The world stops.

The religious community, by the way, is very close. This matter of khessed, of helping your fellow man, is very strong in religious communities; it is a commandment, like eating kosher and keeping the Sabbath.

There might also be an element of learned blankness. That is, going to Rabbis to sort out moral dilemmas for you instead of thinking about it yourself. Without religion it often becomes harder to know what is right or wrong and perhaps Aumann does not like that.

There was a period fifteen, twenty years ago when stealing software was considered okay by many people, including many academics. There was an item of software that I needed, and I was wondering whether to steal it or make a copy of which the developers of the software disapprove. Then I said to myself, why do you have to wonder about this? You are a religious person. Go to your rabbi and ask him. I don’t have to worry about these questions because I have a religion that tells me what to do. So I went to my rabbi a holocaust survivor, a very renowned, pious person. I figured he won’t even know what software is will have to explain it to him. Maybe there is a Talmudic rule about this kind of intellectual property not really being property. Whatever he’ll say, I’ll do. I went to him. He said, ask my son-in-law. So I said, no, I am asking you. He said, okay, come back in a few days. I’ll make a long story short. I went back again and again. He didn’t want to give me an answer. Finally I insisted and he said, Okay, if you really want to know, it’s absolutely forbidden to do this, absolutely forbidden. So I ordered the software. In short, you can be a moral person, but morals are often equivocal. In the eighties, copying software was considered moral by many people. The point I am making is that religion at least my religion is a sort of force, a way of making a commitment to conduct yourself in a certain way, which is good for the individual and good for society.

a confirmed Bayesian

I would be careful here. I find it useful to think that to be a 'rationalist' you actually have to be a rationalist. That is, you need to actually practice what you have learnt otherwise you only know about rationality. Aumann knows a lot about Bayesianism, but this doesn't necessarily make him a confirmed Bayesian. It just means that he knows a lot about what a Bayesian agent would be.