What is the physical meaning of enhanced gauge symmetries in string compactifications? For some very special cases of string compactification backgrounds, we get mysterious enhanced gauge symmetries. What is their physical meaning, and why do they arise? Is there any geometrical interpretation for the enhanced gauge symmetries?
 A: The enhanced gauge groups look fascinating, indeed.
The enhanced continuous gauge symmetry arise because in many cases, one may prove that for particular values of the radii and Wilson lines in various compactifications etc., there exist additional massless spin-one states in the single-string spectrum.
Because string theory produces consistent physics in spacetime for any value of the radii and Wilson lines, and because Yang-Mills theory is the only consistent way to make spin-one fields interact, it follows that spacetime dynamics has to include the enhanced Yang-Mills symmetry.
Moving away from the enhanced-symmetry point (in the space of the radii and Wilson lines) is equivalent to the Higgs mechanism. The scalar fields that measure the deviation (of the radii and Wilson lines) from the enhanced-symmetry points may be interpreted as Higgs fields that spontaneously break the enhanced symmetry.
The emergence of Yang-Mills symmetry was proved from the consistency of string theory which may look like too powerful and abstract a method. However, the emergence may also be showed very explicitly.
The new massless gauge-bosons - analogous to W-bosons that become massless at some point, enhancing $U(1)$ to $SU(2)$, if I mention the simplest example - are obtained by a "purely stringy effect", namely from strings that carry a nonzero winding number. That's why these states can't be seen in the "effective higher-dimensional field theory": the latter has no wound strings.
If those new string states are spin-one states and they are charged under the $U(1)$ and similar "generic" groups, it follows that the physics has to be Yang-Mills theory. But it may be useful to see why there's at least the "global enhanced symmetry group" such as $SU(2)$ at the enhanced point.
The appearance of the global symmetry may be seen by the method of "current algebras". Instead of states of a closed string, it's very useful to switch, by the state-operator correspondence, to the local operators on the world sheet. Much like the total $U(1)$ generator may be written as an integral of a charge density over the string, the remaining $SU(2)$ or other generators may also be written as integrals of a current (its temporal component) over the world sheet.
And indeed, it may be proved that at the enhanced point, the currents have the right OPEs (operator product expansions)
$$ j^a(z) j^b(0) \sim \frac{ k^{ab}}{z^2} +\frac{ic^{ab}_{\,\,\,\,\,c}}{z}j^c(0) $$
Because of the general CFT calculus, these OPEs know all about the commutators of the resulting charges, so one may prove that the charges obtained as integrals of those currents have the right commutation relations. One also has to show that they commute with the Hamiltonian - they're conserved i.e. they're symmetries.
The simplest and most important triplet of the currents that one should understand are those that generate the $SU(2)$ algebra in a circular compactification on the self-dual radius:
$$\partial X, \quad :\exp(iX):, \quad :\exp(-iX):$$
I omit constants, both prefactors and those in the exponent. The $\partial X$ current for a compactified dimension $X$ has the OPEs with others where one simply differentiate with respect to $X$. So it shows that the first current is that of $J_z$ and the other two are $J^\pm$. The most nontrivial commutator or OPE that has to come out right is the OPE of the last two exponentials, but it indeed produces the right result when it should. (Quantum subtleties are critical to calculate the OPE: the commutator is not just the Poisson bracket in this case!)
It is typical that the enhanced symmetries separately arise from left-movers (holomorphic $z$ dependence) and right-movers (antiholomorphic $\bar z$ dependence) - two CFTs that just happen to be combined.
From a more overall perspective, one can't say that the enhanced symmetry is an "unnatural" point and the generic one is "natural". Any point of the moduli space is equally consistent as any other. It sometimes doesn't even make sense to ask what's the "maximum gauge group" that may be broken by the Higgs mechanism.
This has fascinating consequences. 
A simple example. Take the $SO(32)$ heterotic string: the rank of the gauge group is 16 and the dimension is 496. It looks pretty big. Compactify it on a circle. Turn on generic Wilson lines. It will break the gauge group to $U(1)^{16}$, aside from the additional $U(1)$ from the circle itself (one $U(1)$ from the Kaluza-Klein $U(1)$, another $U(1)$ from a reduction of the B-field along the circle).
At an enhanced point, i.e. for proper values of the radius and the Wilson lines, the gauge group becomes $E_8 \times E_8$ - times the same $U(1)^2$. You may decompactify the dimensions differently: if you allow compactifications on a circle, one may interpolate between the $SO(32)$ and $E_8\times E_8$ heterotic string theories by totally consistent theories! Now, notice that $E_8\times E_8$ has rank 16 and dimension 496, much like $SO(32)$. 
In field theory, you would think that there's always the "biggest gauge group" that may perhaps be broken by the Higgs mechanism. In string theory, however, there may be two totally equally big (both rank and dimension), totally different "initial" gauge groups that may be broken to a more generic gauge group. It makes absolutely no sense to ask which of the two heterotic string vacua is more unbroken, more symmetric, or more fundamental. They're clearly equally unbroken, equally symmetric, equally fundamental.
For more general cases of enhanced symmetry groups, one has to learn the Dynkin diagrams, weights, roots, Cartan subalgebras etc., but the derivations are de facto just straightforward generalizations of the simplest case of the $SU(2)$ enhanced symmetry. And it is very typical for the CFTs - that are used as world sheet CFTs here in string theory - have several totally equivalent descriptions that use totally different degrees of freedom. For example, 1 boson is equivalent to 2 fermions when the allowed projections and boundary conditions of both kinds of fields are properly adjusted.
These were gauge symmetries coming from bulk closed strings. Non-Abelian gauge symmetries in string theory also arise on stacks of D-branes - from open strings - and on singularities - the ADE singularities in M-theory (and their F-theory generalizations) are the key example. In M-theory (and type IIA), the enhanced non-Abelian gauge symmetries on the singularities arise from M2-branes (or D2-branes) wrapped on 2-cycles that shrink to zero size. The gauge bosons are literally tiny membranes, much like they're wound strings or open strings in the other pictures. All these pictures how non-Abelian gauge symmetries may emerge in string theory may be related by various dualities. 
Of course, I skipped one more method how a non-Abelian group emerges - the $E_8$ group (and the vector supermultiplet) that has to live on all the boundaries of the 11-dimensional spacetime of M-theory.
A: If I had to take a guess I'd say these gauge symmetries arise in the same way they do in condensed matter systems as an effective macroscopic description. If you know how this happens then skip ahead to the section marked with (*****).
Consider a lattice, for instance the honeycomb lattice, perfectly uniform. Consider the case when the lattice is made of carbon atoms at the vertices with $sp^3$ hybridized orbitals forming covalent bonds. Carbon has 4 valence electrons. Three of these are used up in the 3 planar $sp^2$ ($1s+p_x+p_y$)bonds. The fourth electron lives in the $p_z$ orbital which can hold a maximum of two electrons with opposite spins. Each vertex therefore has one electron when it can hold two. The Fermi energy of graphene therefore lies in the center of its conduction band. This setup is referred to as half-filling.
Then, as has been discussed in a previous question, in the limit that lattice size goes to $\infty$ the effective many-body description of the electron gas is given by the massless Dirac equation. The quasiparticles (distinct from the electrons themselves) are therefore massless fermions.
Any real graphene lattice, like any other condensed matter system, contains defects and disorder. These can be of various kinds involving doping with other substances to change graphene's electronic properties. We are concerned with lattice defects primarily those where instead of a hexagon we find a pentagon or heptagon. An example is the Stone-Wales defect where rotation of an edge creates two heptagons and two pentagons where there were four hexagons previously.
The presence of defects will obviously modify the long-range effective theory. There are various ways one might imagine generalizing the situation given that initially we have only the massless Dirac equation to work with. An obvious possibility is that the dynamics gets modified in such a way that a gauge field is also required. Then we have massless fermions but now there is a (effective) gauge field which mediates interactions between these.
As is shown in this paper (GGV) by Gonzales, Guinea and Vozmediano, this is indeed the case and rather surprisingly the emergent gauge field in this example is $SU(2)$!
(*****) Returning to your original question, I would say that that when you take a 10 spacetime and compactify 6 of its dimensions, you can do so such that different regions of the 10D manifold get compactified differently. We are left with a 4D spacetime, with these compactified objects behaving as defects in this (effective) background. The enhanced gauge symmetries then follow.
Hopefully I didn't completely misunderstand your question!
A: Consider a compactification of M-theory on a $G_2$ holonomy manifold $X$, which in some local neighborhood can be represented as $X=(C^2/Z_N)\times Q$, where $Q\in X$ is a supersymmetric (associative) 3-cycle. The desingularization limit of the orbifold $C^2/Z_N$ is the so-called ALE (asymptotically locally Euclidean) space containing an entire $SU(N)$ root lattice of supersymmetric two-cycles, i.e. one can choose a basis for the two-cycles inside $C^2/Z_N$ such that their intersection matrix is minus the Cartan matrix of the $SU(N)$ Lie algebra. In the smooth limit, i.e. when the two-cycles have a finite volume, one obtains a $U(1)^N$ gauge theory where the N Abelian vector fields come from the first sum in the dimensional reduction of the 11-D supergravity 3-form $C_3=\sum_{i=1}^NA^i_1\wedge \phi_i + axions$, where $\phi_i\in H^2(X,Z)$ are the corresponding basis harmonic two-forms. Now consider the singular limit when all these two-cycles were shrunk to zero size. In this case, there will be new massless gauge degrees of freedom coming from the membranes wrapping the collapsed two-cycles in $C^2/Z_N$ (when a membrane wraps a two-cycle, the mass of the corresponding state is proportional to the volume of the cycle, when the volume shrinks to zero the worldvolume of the membrane effectively turns into a worldline of a massless particle) and when combined with the N vector fields from the KK reduction of the 3-form they fill out the adjoint representation of $SU(N)$. Thus, M-theory compactified on such a space would give rise to a 7-dimensional non-Abelian $SU(N)$ gauge theory living in $R^{3,1}\times Q$. Upon further reduction to 4 dimensions, the volume of supersymmetric tree-cycle $Q$ becomes identified with the inverse of the gauge coupling of the 4-dimensional $SU(N)$ gauge theory (at the KK scale). So, from this perspective the value of the unified gauge coupling in a SUSY GUT, e.g. $SU(5)$ GUT, is directly related to the volume of the three-cycle $Q$ supporting the gauge degrees of freedom in the visible sector $\alpha^{-1}_{GUT}\approx 25=Vol(Q)/{l_M^3}$.
