Motivation behind the neighbor-joining distance matrix recomputation
NJ as with all clustering methods there isn't an explanation, It is a hierarchical clustering method but there are many algorithms to cluster a distance matrix. NJ is closely associated with the concept behind UPGMA (unweighted pair group mean algorithm)

The key difference with UPGMA is that after joining two taxa by NJ the reconstructed genetic distance between the node and one taxa versus the other taxa is not assumed to be equal, i.e. the rate of mutation is heterogeneous. All other clustering algorithms I know assume a "global molecular clock" if that makes any sense, the mutation is constant against time. NJ removes that assumption and thats why we like it.