I found the best introduction to be simulation. The euclidean version of Anderson localization can be stated as follows--- a collection of particles diffuse on a 2-d or 3-d grid. Each particle has a probability per unit time of replicating, this probability is a random function of the position. For convenience of simulation, whenever a particle replicates, another particle is deleted, so that the total number stays fixed.
Then you can ask whether the cloud of particles spreads out to fill the whole space, or whether the cloud is localized in a little region forever. The naive pre-Anderson intution is that the particles will diffuse away from their initial position, but this intuition is totally wrong. A simple simulation shows you that in 3d, you get localization after a certain amount of randomness strength, a large cloud of particles never moves.
In this formulation, there is a direct analogy with a biological phenomenon--- the quasispecies of virus evolution. You can think of each particle as a virus, it's position is its DNA sequence, and the replication rate is the fitness of this sequence. No two individuals of the cloud of viral particles are likely to have the exact same sequence, the diffusion rate is the mutation rate, and there is likely to be one mutation in each replication (viral polymerase is crappy). The dimensionality of the space in this case is effectively infinite, but this is not a qualitative difference from 3d, three dimensions and infinite dimensions both have an Anderson transition.
The resulting localization of sequences is called the formation of an "Eigen quasispecies", and before modern sequencing, it was a controversial idea. The quasispecies of viral evolution is just the Anderson localization in analytic continuation to diffusion.
The biological analogy allows you to see how you can make the localized clump move around--- you can use a process of "bottlenecking". This means you artificially reduce the population to one (randomly chosen) particle, and let it fill out it's own cloud. The cloud is localized again, but likely with a somewhat different center. By repeating the process, you can see that evolution can proceed in viruses whenever there is an infection of a new individual by one or a few viral particles.
This simulation allows you to figure out all the important properties of the Anderson transition by yourself without consulting literature. The methods Anderson uses are also clarified, because Anderson is doing everything in perturbation theory using real-time quantum mechanics, not diffusion, where the process is less intuitive, because it is in amplitude, not in probability.