Motivation behind the neighbor-joining distance matrix recomputation In each iteration of the neighbour-joining method of phylogenetic trees construction, after joining the nearest neighbours the (additive) distance matrix $D_{m \times m}$ is recomputed as
$$Q_{ij} = (m-2)d_{ij} - \Bigg( \sum_{k \neq i} d_{ik} \Bigg) - \Bigg( \sum_{k \neq j} d_{jk} \Bigg)$$
As is stated in numerous sourced that I found, it prevents the joining of non-direct neighbours, but am looking for a more detailed explanation and motivation.
Sources: Wikipedia
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.