Myhill-Nerode equivalence classes of $\{1^n0^n\}$
I think it makes sense to answer your subquestions out of order.




  Why they don't write $[1^n0^m] = \{1^n 0^m | n,m \in N^+\}$


They're supposed to be equivalence classes. $[1^n0^m]$ would break up lots of classes, and duplicate lots of the states from line 3.




  What does the Exponent $n+i-1$ mean?


Firstly, $n+i-1 \ge i$, so there are at least as many $1$s as $0$s.

Secondly, $n$ is the parameter which defines the class.

So $\left[1^{\mathrm{l}}0\right]_{\equiv \mathrm{A}}$ is the set of states which need $0^{l-1}$ to reach the accepting state.