Biological meaning of Lotka-Volterra Jacobian matrices
The Jacobian tells how the system changes along different state variables (which can be, for instance, the concentrations of the predator and the prey).

The Jacobian matrix by itself doesn't give you a lot of intuitive information. However, the eigenvalues of the Jacobian matrix at the equilibrium point tell you the nature of the steady state. For example if all the eigenvalues are negative then the system is stable at that point and the equilibrium point is called a stable node. If some of them are positive and some are negative then the system is metastable at that point and the point is called a saddle point. If all eigenvalues are positive then the point is unstable. If the eigenvalues have an imaginary part then the system exhibits oscillations. All these properties are local i.e. they just tell you the behaviour of the system near the equilibrium point.

I hope I am clear. This is not in a strict sense, a biological question. If you want to know more then read a book on nonlinear dynamics. Non-linear dynamics and Chaos by Steven Strogatz is a good place to start.



EDIT

When the system is locally linear (linearized at a given point) then the system dynamics at that point can be described as 
$$\frac{dx}{dt}=J.x$$

Where $x$ is the state vector (a vector of the variables in the system) and $J$ is the Jacobian matrix. In this situation, the signs will denote the effect of a variable on the other variable. However, I am not sure (I feel it is highly unlikely) if this can be used to make any inferences about the food web.