A full introduction to the DMRG algorithm definitely does not fit here, and you can find many well-written introductory materials online.
DMRG has been applied to simulate perturbed toric code model, which is the simplest example of string-net model, see http://arxiv.org/pdf/1205.4289.pdf. Generally speaking, DMRG for 2D spin models is indeed much more difficult to implement, however there have been remarkable progress in recent years along this direction. Usually in this type of simulations spin models are placed on a long cylinder. The complexity grows exponentially with the circumference of the cylinder, while the length can be very long, so in a sense this is a quasi-one-dimensional geometry.
I suppose what you looked for is some kind of algorithms that take advantage of the tensor network structure in 2D. A lot of effort has been put into the development of such algorithms, but to date it is fair to say that nothing as useful as DMRG in 1D has come out yet.