The matrix above depicts the signs of the partial change of a function with respect to each species. Because there are 4 rows and columns, this means there are 4 species. Each row represents a partial change in each of the 4 differential equations representing the rates of change for each species with respect to each species, which is each column. To rewrite the Jacobian for species 1–4 ($N_1, N_2, N_3, N_4)$:
$$\mathbf{J} = \left[\begin{smallmatrix}
\frac{\partial F_1}{\partial N_1} & \frac{\partial F_1}{\partial N_2} & \frac{\partial F_1}{\partial N_3} &\frac{\partial F_1}{\partial N_4} \\
\frac{\partial F_2}{\partial N_1} & \frac{\partial F_2}{\partial N_2} & \frac{\partial F_2}{\partial N_3} &\frac{\partial F_2}{\partial N_4} \\
\frac{\partial F_3}{\partial N_1} & \frac{\partial F_3}{\partial N_2} & \frac{\partial F_3}{\partial N_3} &\frac{\partial F_3}{\partial N_4} \\
\frac{\partial F_4}{\partial N_1} & \frac{\partial F_4}{\partial N_2} & \frac{\partial F_4}{\partial N_3} &\frac{\partial F_4}{\partial N_4} \\
\end{smallmatrix}\right]
$$
The signs above correspond the effect of species on each other's rates of change. For a first example, the first row and column is effect of species 1 on its own rate of change. It's negative, meaning that it's rate of change decreases with increasing density (crowding). A second example is the first row and second column—species 2 has a negative effect on the growth of species 1. In a food web context, this means that species 2 consumes species 1. A third example is the first row and third column—there is no effect of species 3 on species 1. Lastly, the second row and first column has a positive sign, meaning that species 1 has a positive effect on the growth rate of species two. In a food web context, that means that species 2 consumes species 1.
Notice that in food webs, for each positive effect, there is a corresponding negative effect and vice versa (i.e., species grow from eating another species, and species shrink from being eaten).
In sum, (1) species 1 is the only species who is affecting by crowding, (2) species 1 is consumed by species 2, (3) species 2 is consumed by species 3, (4) species 3 is consumed by species 4, and (5) no other species interact.