Here is the way I would try to explain Loop Quantum Gravity to my grand mother. Loop Quantum Gravity is a quantum theory. It has a Hilbert space, observables and transition amplitudes. All these are well defined. Like all quantum theories, it has a classical limit. The conjecture (not proven, but for which there are many elements of evidence), is that the classical limit is standard General Relativity. Therefore the "low energy effective action" is just that of General Relativity.
The main idea of the theory is to build the quantum theory, namely the Hilbert space, operators and transition amplitudes, without expanding the fields around a reference metric (Minkowski or else), but keeping the operator associated to the metric itself. The concrete steps to write the theory are just writing the Hilbert space, the operators and the expression for the transition amplitudes. This takes only a page of math.
The result of the theory are of three kind. First, the operators that describe geometry are well defined and their spectrum can be computed. As always in quantum theory, this can be used to predict the "quantization", namely the discreteness, of certain quantities. The calculation can be done, and area and volume are discrete. therefore the theory predicts a granular space. This is just a straightforward consequence of quantum theory and the kinematics of GR.
Second, it is easy to see that in the transition amplitudes there are never ultraviolet divergences, and this is pretty good.
Then there are more "concrete" results. Two main ones: the application to cosmology, that "predicts" that there was big bang, but only a bounce: And the Black Hole entropy calculation, which is nice, but not entirely satisfactory yet.
Does this describe nature? We do not know...
carlo rovelli
So what are spin-networks? Briefly, they are graphs with representations ("spins") of some gauge group (generally SU(2) or SL(2,C) in LQG) living on each edge. At each non-trivial vertex, one has three or more edges meeting up. What is the simplest purpose of the intertwiner? It is to ensure that angular momentum is conserved at each vertex. For the case of four-valent edge we have four spins: $(j_1,j_2,j_3,j_4)$. There is a simple visual picture of the intertwiner in this case.
Picture a tetrahedron enclosing the given vertex, such that each edge pierces precisely one face of the tetrahedron. Now, the natural prescription for what happens when a surface is punctured by a spin is to associate the Casimir of that spin $ \mathbf{J}^2 $ with the puncture. The Casimir for spin $j$ has eigenvalues $ j (j+1) $. You can also see these as energy eigenvalues for the quantum rotor model. These eigenvalues are identified with the area associated with a puncture.
In order for the said edges and vertices to correspond to a consistent geometry it is important that certain constraints be satisfied. For instance, for a triangle we require that the edge lengths satisfy the triangle inequality $ a + b \lt c $ and the angles should add up to $ \angle a + \angle b + \angle c = \kappa \pi$, with $\kappa = 1$ if the triangle is embedded in a flat space and $\kappa \ne 1$ denoting the deviation of the space from zero curvature (positively or negatively curved).
In a similar manner, for a classical tetrahedron, now it is the sums of the areas of the faces which should satisfy "closure" constraints. For a quantum tetrahedron these constraints translate into relations between the operators $j_i$ which endow the faces with area.
Now for a triangle giving its three edge lengths $(a,b,c)$ completely fixes the angles and there is no more freedom. However, specifying all four areas of a tetrahedron does not fix all the freedom. The tetrahedron can still be bent and distorted in ways that preserve the closure constraints (not so for a triangle!). These are the physical degrees of freedom that an intertwiner possesses - the various shapes that are consistent with a tetrahedron with face areas given by the spins, or more generally a polyhedron for n-valent edges.
Some of the key players in this arena include, among others, Laurent Friedel, Eugenio Bianchi, E. Magliaro, C. Perini, F. Conrady, J. Engle, Rovelli, R. Pereira, K. Krasnov and Etera Livine.
I hope this provides some intuition for these structures. Also, I should add, that at present I am working on a review article on LQG for and by "the bewildered". I reserve the right to use any or all of the contents of my answers to this and other questions on physics.se in said work, with proper acknowledgements to all who contribute with questions and comments. This legalese is necessary so nobody comes after me with a bullsh*t plagiarism charge when my article does appear :P
Best Answer
The original construction of the Ashtekar-Lewandowski space only depends on the fact tha gauge group is compact. As long as you stay at the kinematical level, and don't bother about the dynamics, any gauge group can be implemented if it is compact.
The gauge group of the standard model, in particular, can easily be incorporated by developing spin networks, not for $SU(2)$ but for $SU(2) \times G_\textrm{Standard} \simeq SU(2) \times (SU(3)_C \times SU(2)_L \times U(1)_Y)$. In practice, it means that the edges of the spin network will not carry spins but rather (unitary) representations of the full gauge group, which can be labelled by a spin (for the gravitationnal gauge group) but also labels for the representations of $SU(3)$, $SU(2)$ and $U(1)$ from the Standard Model.
Now, for matter, the subject is less clear to me. It seems that the Hilbert space is well-defined for fermions but, at the dynamical level, we don't know how to do this without issues regarding chirality which translates into fermion doubling problems (http://arxiv.org/abs/1507.01232) though some people think it might be possible to do (http://arxiv.org/abs/1506.08794).
For scalar matter (like the Higgs), for a long time, compactification was the easiest way to deal with the definition of the problem (the scalar field takes value in $U(1)$ for instance) as was done in Thiemann original work (the whole thing is developed here - if you ever find the courage to read it).
More recent developments used Bohr compactification. See for instance : http://arxiv.org/abs/gr-qc/0211012 though the consequence of this doesn't seem clear to me.
All these problems though can be classified into two kinds:
At present, I would say there are some pretty convincing ways of incorporating matter into LQG. The main trick though, as always with LQG, is the dynamics.