Chapter $15$ of Davenport's Multiplicative Number Theory writes that
Assuming for simplicity that $T$ (which we suppose to be large) does not coincide with the ordinate of a zero, we have $$2\pi N(T)=\Delta_R \operatorname{arg} \xi(s)$$ where $R$ is the rectangle in the $s$ plane with vertices at $2$, $2 + iT$, $-1 + iT$, $-1$ described in the positive sense.
I have the following questions
1-What is $\Delta_R$ here. I tried searching it in previous chapters but could not understand what is this notation.
2-How this equation has been derived.
In the formula $$ 2\pi N(T)=\Delta_R {\rm arg} \xi(s) $$ the term on the RHS stands for the increment of the argument of $\xi(s)$ along the boundary of the rectangle $R$, i.e., the change in the argument along the path of the boundary of $R$.
For details see for example here, page $436$. How this is connected to the argument principle, see here, or here, page $10$.