How does one cope with an unfalsifiable (and hence: unprovable) belief? The simplest way is to accept it as an axiom. Ultimately, belief in God requires just that - an act of faith.
What good is it though? Some pretty obvious suggestions follow:
1. Ontological stop-gap
While science - by which I mean here the practice of constructing predictive theories that are then validated against empirical data - offers the best way we know of learning how the universe we live in behaves, there are clearly questions science as we know it is incapable of answering.
For example: how did it all begin?
Given that we have no concievable way of peering "outside" the universe, to the time when there was none - and yet the preponderance of scientific evidence strongly suggests that there was a beginning (what we call the Big Bang) - the act of creation by a personal God is as good an explanation as any.
We have no way of telling if that was the case (unfalsifiable), but it is an answer some people are willing to accept.
Another example would be why does the universe behave like it does?
Here we briefly touch on the anthropic principle and again God is as good an explanation as any. Notice that whether we accept a theistic explanation or not has no bearing on our ability to apply the resulting principles. Newton saw the laws of physics that he studied and codified as being instituted by the Divine - and that's a reasonable explanation for a first cause. It is unlikely that we will be able to determine with any certainty the circumstances that caused physics to be the way it is. Nevertheless, we are able to use the "laws" to our advantage, regardless of accepted origin.
2. Ethical foundation
A moral realist might well ponder whence the principles that govern ehtical behaviour and again, God is a reasonable explanation. Indeed, I would posit that religion is - at its core - an ethical proposition, rather than a purely ontological one. We cannot understand the Divine - we are told (which may well be true) - and we likely cannot fully understand the universe in which we live, but we can follow the ethical precepts that stem from revelation.
The idea that there may be no objective right or wrong is disturbing to many people. The world as "is" cannot inform us of what we "ought", for the very simple reason that in order for there to be an "ought" there must necessarily be an "ought not" - a path into the future we can, but must not take. Without choice, the "ought" becomes merely a "will".
Having accepted God as an ontological stop-gap - an answer to the unanswerable questions - it is not big stretch to accept that the same Divinity that set out physical laws also set out moral ones.
Given that ethics cannot be derived from empirical data, for reasons given above (the universe may tell us what results our actions may have, but not how we should judge these results and therefore the actions leading to them), such an ethical stop-gap is pretty much the best answer we can hope for.
How can belief be taught to one not already favourably inclined?
Most, if not all, evangelical work consists in improving the potential convert's inclination to the answers being provided.
Granted, most proselytism in history has taken place among people who were already favourably inclined towards this type of answer. It is a smaller step to accept you have believed in false gods, than to go from a "I have no need for that hypothesis" mindset to finding such a need.
Ultimately, however, the atheist must either accept that they have no answer to the kind of questions that God is the answer to - content perhaps with the view that neither does anyone else - or they will find that some agency outside this world must necessarily be introduced. God is perhaps the most palatable such agent, being both human-facing and favourably inclined towards humanity.
How can competing beliefs co-exist?
This has partially been touched upon in the previous section: all religions offer a similar type of answer and thus have an easier time co-existing with one another than with a worldview that rejects this type of answer outright. We have many historical accounts of competing religions existing in the same place - sometimes even relatively peacefully - but atheism has been hated by just about every religious society.
Much like relativity and quantum physics offer two views of the world that aren't quite compatible, two religions can debate which of them is closer to the "actual nature" of God and His commandments. However, neither of them will readily accept a proposition that not only are they both incorrect, but that the kind of answers they provide can never be correct.
On a personal level, one who is inclined to accept God as an answer to unanswerable questions may not be content with any specific interpretation of this answer (in ontological or ethical terms), but may nevertheless believe that the correct answer is out there, somewhere - perhaps diluted amongst the competing messages. One's personal quest for faith becomes, therefore, an attempt to distill Divine wisdom from the dregs of imperfect human understanding and teaching.
Addendum: Unfalsifiable beliefs as zero-value propositions
I feel this answer would be incomplete without touching on the epistemic considerations of an unfalsifiable belief. I choose to call such beliefs "zero-value", by which I mean that their adoption or rejection has no effect on the rest of our knowledge.
It doesn't matter whether we believe God set down the laws or physics or not - we can make predicitons based on them based on what we discover them to be, rather than their source.
A belief (hypothesis, theory) has positive epistemic value if it allows us to make (reasonably) accurate predictions we would not have been able to make, were we to leave this belief out of our predictive model.
Conversely, a belief has negative epistemic value if its adoption causes the accuracy of our predictions to decrease.
Aside: It is worth noting that a belief may have both positive and negative value, depending on the domain. Newton's laws are perfectly useful in everyday life (positive epistemic value), but acquire negative epistemic value if we try to apply them outside their proper domain. Neither the relativistic nor the quantum-scale universe obeys Newton's laws.
A "zero-value" belief is perfectly neutral: it neither adds to our knowledge (as evidenced in predictive ability) nor deducts from it. Russell's teapot is an example of such a "zero-value" proposition: it's existence (or not) has no bearing upon our predictive abilities - as long as we do not try to falsify it. Arguably, because it is a falsifiable belief (we can scour space until we've found it or checked every last cubic inch), it has at least the possibility of having a non-zero epistemic value. More contrived - but airtight - examples are trivial to come up with (undetectable dancing unicorns and such).
The key point is that we can add or subtract any number of such "zero-value" propositions to and from our sum of knowledge and they will not affect the rest of it in any way - as far as ontology goes.
Ethically, the matter is not as clear cut, because ethics are not subject to predictive evaluation. Ethical systems are themselves unfalsifiable.
Whether multiplication of zero-value propositions in our system of knowledge is necessary or a sin against Occam's Razor is a matter of personal choice.