Mutations in DNA or RNA sequences do not necessarily result in significant changes in the functions of the proteins they encode (or in the case of RNA ribozymes, ribozyme function ). This is because, for a variety of reasons, the change in DNA/RNA sequence may not significantly alter the structure and function of the ribozyme/protein (the function that contributes to the fitness of the organism). If two alleles of a gene with different sequence have equivalent function (again with regard to the function of the gene that provides fitness) the differences between them are selectively neutral.
The neutral network consists of every variation in the sequence of the gene that can be reached from another sequence (of equivalent fitness) by a single base pair mutation (these aren't the only type of mutation, possible, but they are common and more likely to be neutral than more drastic changes). So for example (using an unrealistically short sequence), if I had a gene with the sequence 0 AACAATGCTGACTGA, and alleles 1 AACAATGCTGACTAA, 2 AACAATGCTGACAGA, 3 AACAATGCTGACTGG with equivalent function, 1, 2, and 3 would be directly linked to 0 in the neutral network as they all differ from 0 by a single base pair. The sequence 4 GACAATGCTGACTAA if of equivalent function would also be in the network as it differs by one base pair from sequence 1.
Each gene in the network being mutually accessible through mutations means that I could start out with any of the above sequences and reach any of the others through a series of single base pair mutations that traverse the network (i.e. I could mutate 0 into 4 by first mutating to 1, another network member, then to 4).
The genotypic space includes the sequences of all extant alleles of the gene. This paper suggests that there is a tendency for these sequences to form these networks and especially to evolve towards highly interconnected networks. This is the structural information the author refers to. It should be easy to imagine how this might confer robustness to deleterious mutation. Let's think about our sequences from above, imagine if every sequence was only linked to two other selectively neutral sequences forming a long chain. The sequence AACAATGCTGACTGA is 15 base pairs long with 45 possible single mutations, if only two of those are selectively neutral, the chance of mutating to a neutral network member is low, and even if you do hit on a neutral mutation, your chance of hitting another neutral mutation next time (reverting back or moving on down the chain) is still only 2/45. Now imagine that the neutral network is highly interconnected, with sequence 0 linked to sequences 1, 2, 3 and several other neutral 1 base pair changes, imagine those sequences are linked through each other by other neutral intermediates (being 2 base pair mutations away from 0). Not only is the chance of a single mutation being neutral higher, but subsequent mutations (remember that these include reversions from prior mutations) are more likely to stay within the network (you're more likely to come back around to 0.
Would you have more long-term success walking around a fitness plateau or walking across a fitness rope suspended over the abyss?