Why is it important that physics consists of unitary phenomena? AFAIK all known quantum laws are unitary (except the collapse postulate, which is dubious anyway).
Why is it important that the laws be unitary?
 A: Because symmetry is an important principle in physics, and it dictates unitarity (but see the end of my answer for an additional possibility). Also see my answer here on unitary time evolution.
Symmetry here means symmetry in physical laws when going from one point of view to another. As far as physics is concerned, us humans observe/measure the results of physical experiments.  If you perform an experiment in two different ways, but still obtain the same result of measurement, we say that it constitutes a symmetry between the two ways.
Symmetries are the cornerstone of physics. If there were no symmetries, our universe would be very different since any two different points of view would give different results of measurement; a chaotic place to be indeed. It turns out that we are not so unlucky as there certainly are symmetries in play in our universe. Historically, it took us time to discover the presence and to appreciate the variety, pervasiveness and usefulness of symmetries in our physical laws. Variety, because there are not one but many symmetries. Pervasiveness, because seemingly different theories follow the same symmetry principles, thereby allowing us to unify them in one general framework (e.g. electricity + magnetism = electromagnetism). Usefulness, because the presence of symmetries (along with the above-mentioned unifying general framework) allows us to construct theories that are predictive: we can calculate, using our theory, what the result of an experiment would be (and then later compare it with actual measurement in order to verify the theory's validity). The presence of symmetries, in the first place, allows us to engineer a predictive theory; without symmetries, we would actually have to perform an experiment to know what its result would be, since the absence of a symmetric structure in our physical laws would preclude us from calculating anything a priori.
Now, symmetries are important and we know we have them in our universe, but then what? How do we add (or base upon them, rather) them to our theory, before actually using its consequences? That's Wigner's theorem, which says that every symmetry transformation that acts on physical states (rays), can be represented in the Hilbert space of physical states by:
a unitary and linear operator
$\qquad$ or
an anti-unitary and anti-linear operator.
(So note that you can have anti-unitary operators as well, eg. time reversal.) In the context of quantum mechanics, conservation of transition probabilities, under a transformation, constitutes a symmetry.
A: Unitary is indeed connected to symmetry; it is therefore an important organizing principle which we should look for and expect to find at some level. This does not imply that we can claim to know it will be found exactly, or always applying to every process. Maybe it will, maybe it won't. In this respect it is like parity conservation: an important concept, but not necessarily the whole story. However, unlike parity, we have, at present, no confident way to proceed with the question "if not unitary, then what?" We have only some tentative ideas of what to look for or how it might go. The idea of a stochastic contribution to fundamental dynamics is not completely ruled out. It is simply that one should not give up unitarity lightly, because it is closely connected to energy conservation and information conservation. One should require a non-unitary model to show good credentials.  
A: Here is another way to look at this. 
You can consider unitarity as a requirement that in any process described by your model, the sum of the probabilities of all the different pathways that the process can take must be equal to one. In a crude sense (and I invite the experts to correct me if my interpretation of this is wrong) if that probability sum is less than one, it means something that should have happened did not and if the sum is greater than one, something that shouldn't have happened did. In either case this indicates that your model doesn't correctly describe the world- either there's an unphysical flaw in your model or in the manner it furnishes probabilities.
