Length of an Edit Script and the Number of Deletions
This is a straightforward consequence of the fact that the edit script only contains insert and delete operations.  It follows from a little bit of basic arithmetic -- it's nothing especially deep.

Consider a script that contains $I$ insert operations and $P$ delete operations.  Then the total number of operations in the edit script is $I+P$.  If we're given that the total number of operations is $D$, we know $D=I+P$.  Also we know $M+I-P=N$ (since each insert operation increases the length of the document, and each delete decreases it).  Solving these two simultaneous equations yields the relationship you list.