Maybe I am oversimplifying but these answers are read also by others with similar concerns so I will start simply.
In classical physics it is easy to measure the cross section of a target. We take a tape measure and get its cross section from the geometry. For objects at a distance we use light intelligently and with corresponding geometrical formulas get the cross section.
We could get it by shooting at a small inaccessible target and counting the hits from the miss and the geometry of bullets and space arrive at a cross section of the target in a messy way ( light is much better).
When we reach the elementary particle sizes of nanometers and less, we enter the quantum mechanical regime and the only tools we have really are in shooting particles against each other and looking at the outgoing particles. Given the quantum mechanical probability formulation the counting of hits and misses gave a "cross section" in centimeters-squared, the same concept as in classical mechanics, but this time dominated by the probabilistic nature of the beast. The formulae used were solutions of quantum mechanical equations and these lead to a formula given in your link as formula 1) . Quantum mechanics said that the scattering could be described in this functional form, where the functions W1 and W2 would be checked by measuring them in scattering experiments. As electrons were the smallest in mass particle that one could manipulate in the lab, electrons were used as beams and protons as targets.
This blog entry might help you, it goes through the history of the terminology.
In the picture above, on the left, experimental data show the form factors (extracted from the scattering rate) as a function of the square of the momentum transfer, q^2. The factors behave according to a simple "dipole formula", which implies that the proton looks like a rather simple object, with a distribution of charge and magnetic moment following an exponential distribution. Note that the dependence on momentum transfer of the form factors implies that the electron is seeing an object with an extended structure, with a well-defined characteristic size.
In this way using the cross section formula we can get a size for the proton which is dependent on the four momentum transfer of the incoming electron, the higher the Q**2 the smaller the proton.
These form factors were tabulated and used to confirm the simple model, that the proton behaves as a type of uniform distribution of matter.
So the data conforms to a simple model, but this does not yet explain much of what the proton and the neutron actually contain. In order to gather more information, higher-energy interactions are required.
This simple picture was modified once higher energies were reached, because it became apparent that, (as in the Rutherford experiment), the electrons were hitting a hard core , a second level of scattering from charges within the proton.
In general, structure functions are mathematical functions entering the cross section calculations which can be measured and tabulated and tested against models.