Implement multi-fragment heuristics for the traveling salesman problem
there are almost no other information regarding this algorithm online
  [...]
  I would really appreciate a pseudo-code, if anyone has ever implemented this algorithm.


I invite you to read my paper "An Empirical Study of the Multi-fragment Tour Construction Algorithm for the Travelling Salesman Problem" appeared in the Proceedings of the 16th International Conference on Hybrid Intelligent Systems (HIS 2016) pp 278-287. DOI: 10.1007/978-3-319-52941-7_28.

You'll find a pseudo-code of the algorithm, which is similar to the "usual definition". It's explained with some examples of a small instance.
The paper also presents briefly other tour construction methods and gives experimental results for the Multi-Fragment algorithm and a comparative study to show the effectiveness of this method.


  If the first description is correct, I'm actually not understanding it
  fully and how could I implement it in practice.


It's correct but maybe a little bit hard to explain. The main idea is to move columns and rows following the criteria such that we can read the constructed tour from the $n−1$ cells contiguous to the diagonal {D[1][0],D[2][1],... D[n][n-1]}.


  In the first description, they mention the n−1 cells contiguous to the diagonal, but of course we have 2 of this n−1 cells, one in the left triangular matrix (currently from the top to the bottom 3,1,5) and one in the upper one (1,7,0). Is it ok to choose one arbitrarily?


The distances matrix in the Traveling Salesman Problem (TSP) is symmetric $(D[i][j]=D[j][i])$, we may consider only the upper/lower triangular portion of the matrix $D$.
There is another variant of the TSP which is the Asymmetric TSP.