[EDITED because of my inattention] I suspect "operations" here means multiplies and inverses only, neglecting subtractions $x_j-x_k$.
Taking a look at the paper, I see Eq. (2.8):
$$ \phi_{N+1}(x) = \prod_{l=0}^N (x-x_l)\,. $$
Then, [Eq. (2.9)]
$$ \phi'_{N+1}(x) = \sum_{k=0}^{N}\prod_{l=0,l\neq k}^N (x-x_l) $$
and so [Eq. (2.10)]
$$ \phi'_{N+1}(x_k) = a_k \,.$$
For each $a_k$, $N$ multiplies are required. Could one interpret the assertion globally, on all $a_j$? Indeed, the paper evaluate $4N^2$ operations for the whole matrix $D$. What if one only counts multiplies or inverses? Back in 1989, it was not uncommon to forget adds and subtractions in complexity. In Eq. 2.12a, they evaluate off-diagonal elements:
$$\frac{a_j}{a_k(x_j-x_k)}$$
with "another $2N^2$ operations". Once the $a_j$ are computed, this seems sound: $1$ product and $1$ divide for each pair of $(j,k)$, $j\neq k$. Then they count $N^2$ for the diagonal elements $$\sum_{l=0,l\neq k}^{N}\frac{1}{x_k-x_l}\,,$$ ie $N$ inverses for each of the diagonal elements, about $N^2$ total.