Ni (xi, yj) = Δ ij means that this point is one and its point is zero, which is a necessary property of interpolation function. Of course, see the numerical analysis textbook for details.

Ni+Nj+Nm= 1 can be understood as follows: U=NiUi+NjUj+NmUm (U stands for displacement in the x direction). If the element moves in the x direction, all points in the element have the same rigid displacement, and of course, the nodes also have the same rigid displacement. It can be concluded that Ni+Nj+Nm= 1.