Why is energy minimized over the string landscape? As understand it, the 4D string landscape is a function that assigns an energy to every possible compactification of the 6 small spatial dimensions.  We expect our universe to lie in a local energy minimum, and if there is a lower minimum at another compactification, then our universe would only be metastable, because we expect that it would eventually quantum tunnel into the lower minimum.
What is the process that prefers compactifications with lower energy?  I can understand why we might expect total energy should be conserved, but not why it should be minimized.  To me, the logical statistical ensemble with which to describe the universe would be the microcanonical ensemble, in which the total energy is arbitrary and not necessarily minimized.
Usually there are two processes that cause a system to lower its energy: (1) a dissipative system (e.g. one with friction or air resistance) continuously loses energy into an external sink, or (2) a system in thermal equilibrium at zero temperature minimizes its energy because it is coupled to a much larger external heat bath which gains so much entropy from its energy that the total entropy is maximized when all the energy leaves the system and goes into the bath.  In both cases, the idea is that the entropically favored situation is the one in which all the energy leaves the system and goes into some external "environment."  But by definition, the universe cannot be coupled to any external environment, so there is nowhere outside of the universe for the energy to go.  So why do we expect low-energy configurations to be any more likely than high-energy ones?
 A: 
As understand it, the 4D string landscape is a function that assigns an energy to every possible compactification of the 6 small spatial dimensions. We expect our universe to lie in a local energy minimum, and if there is a lower minimum at another compactification, then our universe would only be metastable, because we expect that it would eventually quantum tunnel into the lower minimum.

That's a very strange description, and it is important to first get the terminology right: 


*

*Perturbative string theory is defined by a conformal non-linear $\sigma$-model taking values in a ten-dimensional target manifold $M$, on which it couples to background fields $G,B,A,\Phi$, which are the target space metric, the Kalb-Ramond field, an ordinary background gauge field and the dilaton, respectively. Conformal symmetry of the model imposes the vanishing of all $\beta$-function and yields "string equations of motion" for the VEVs of the background fields. Solutions to these equations together with a fixed geometry for the target $M$ are called vacua of string theory, since they correspond to classical solutions of the effective QFT on the target space that approximates string theory at low energy. Most strikingly, the string e.o.m. for the background metric is, to first order, the Einstein equation.

*The landscape of string theory is now the collection of all these vacua, or, more restrictively if we want a nice 4D phenomenology, the collection of all vacua witha  target $M$ that decomposes as $M^{(4)}\times M^{(6)}$ where $M^{(6)}$ is a compact "internal" manifold and $M^{(4)}$ Minkowski, deSitter or anti-deSitter space. There is intrinsically no "function" that assigns energies to any of these vacua.
Pure string theory itself has no notion of dynamics for these vacua - they are not states in its quantum theory, and they don't tunnel. The background in the standard string theory description is arbitary, but fixed, since it is a starting point for string perturbation theory. However, much in analogy to the vacua in usual QFT, one might conceive of these vacua themselves being stable/meta-stable/unstable (e.g. like the zero-VEV vacuum for the Higgs field) and changing dynamically. I don't know whether some more detailed description of such dynamics is known.
However, the "energy function" you are likely referring to is "obvious": All the effective SUGRA theories on some background have a cosmological constant. It's the value of the cosmological constant - possibly together with a function of overall flux of the higher gauge fields - that one could take as the "energy" of the chosen vacuum. Our standard heuristic understanding of general physics then suggests that this be minimized for stable vacua.
Finally, let me remark that "compactification" is a somewhat misleading name. We are not starting from some non-compact space and compactifying it. We are from the outset searching for soutions to the string background equations of motion, and one such solution, if the internal space is compact, is called a "compactification", but there is generally no process involved that would obtain that manifold from a general non-comapct manifold. This can be seen particularly clearly for fluxless backgrounds on whom preservation of SUSY is imposed - the internal space there is a Calabi-Yau manifold, and those are studied in their own right, not obtained by an actual compactification "process".
A: I think your question can be answered without referring to the specifics of String Theory. What is the physical mechanism driving systems to a minimum of their potential energy? It is the principle of least action.
In these considerations of the String Landscape, one looks at the scalar potential of the 4d effective theory, since the values of the scalars (moduli) in the theory will determine key properties of the theory, e.g., the compactification volume, string coupling etc.
Now, we are interested in stable solutions where all the scalar fields have constant values. Their kinetic energy can dissipate through Hubble friction or decay into light, non-scalar degrees of freedom. Note: in a dynamical background spacetime, energy is not conserved! Energy conservation comes from the time translation invariance, which is broken through the expansion of the universe.
In the end, technically, one equates the Lagrangean to the scalar potential and has the action
$$ S = \int \mathrm d^4 x \mathcal L = -V.$$
Now, throughout the study of field theory, the stable points of a theory were always the extrema of the action. There is no reason to assume otherwise in this case, ergo, the stable points are those that extremise $V$.
Final note: since Lagrange we know that extremising the action gives the physical equations of motion. This is postulated and works well in all theories constructed so far. We have not figured out the deeper reason why the action should be maximised (i.e. potential energy minimised), though.
