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
Here is something, which may be aiming a little low...
The main two ways we describe our universe, quantum mechanics and general relativity, contradict each other when applied simultaneously. This seems to point out that the quantum nature of spacetime itself needs to be understood better.
One way to resolve theoretical problems with our current understanding of spacetime is to embed it in a larger theoretical structure, which has powerful underlying symmetries. Those symmetries cannot be too restrictive: they should be enough not only to make the model well-behaved, but also to be consistent with the not-quite-so-symmetric world we see around us.
This way of thinking has led to the theoretical structure of string theory and then M-theory. To study the structure of the theory, it is useful to first concentrate on the most symmetric situations, even though these are the most removed from our world. At first, this led to study of supersymmetric string theories in 10 dimensions (higher dimensional theories are more symmetric - their Lorentz invariance is larger and more restrictive). Later it turned out that those are all secretly related to an even larger and more symmetric structure, dubbed M-theory, which describes all the previously known string theories as well as 11-dimensional supergravity.
The story is not finished, we only have bits and pieces of the underlying structure that is M-theory. But, we do have many indications we are on the right track. As always with deep structures we found side benefits in the form of unexpected applications in mathematics and physics.
One of the unexpected discoveries is that quantum gravity is not all that different from other parts of physics, and sometimes conventional physics can be reformulated in different variables to make it equivalent to a quantum gravitational theory. Using classical and semi-classical gravity calculations then helps us explore conventional physics in regimes otherwise inaccessible. This is the whole subject of holography and its applications.
So, what we seem to have found is, instead of a specific model to describe our universe at short distances (or high energies), a whole new language in which we describe and discuss physics - and not just high energy physics. Where precisely this is going to lead is anyone’s guess.
Now, if your grandmother feels patronized, has more knowledge of physics and would like to ask more specific questions, I can try to add more details. It is a very large subject...
(see also the answers to this similar question)
Best Answer
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