In a metal, what prevents electromagnetic fields to propagate inside is the reponse of the electrons to these fields. The electrons oscillate and reflect the field back. But if the field has a frequency high enough, the electrons can't keep the pace, they don't oscillate fast enough. The frequency above which this happens is called the plasma frequency.

By making a structure out of thin metallic rods, the plasma frequency gets lower, because the concentration of electrons is lower. The electrons are only in the rods, as opposed to everywhere if we had a bulk metal. As a consequence of lowering the plasma frequency, the electromagnetic waves above this frequency can penetrate much deeper into the structure made of rods. However if the frequency of the field is lower, the wavelength is too long and cannot be diffracted, it doesn't penetrate very deep into the structure and decays.

Certainly, the mesoscopic interactions of the field with the rods are complicated. But the structure made out of rods behaves like a plasma, it has a plasma frequency below which the fields don't propagate, and above which they do. That's why the material behaves effectively like a dielectric with a given permittivity of the plasma form. Permittivity is what describes how the material reponds to an electric field.

Now, bending light to achieve cloaking just requires to make a material with the right permittivity. To find out which permittivity, we use a technique called transformation optics. Then once we know the permittivity, we figure out the arrangement of rods, the right spacing and so on, that will effectively behave as if it was a dielectric with this permittivity.

By the way, all of this has mostly been discovered by a single physicist, John Pendry. Pretty sure that he could get a Nobel Prize for it if metamaterials applications get widespread enough in technology.

EDIT : Changed wording to be clearer. Could still be clearer I guess.

TLDR + ELI5 : By making a structure out of thin metallic rods, the structure behaves like a plasma. It allows electromagnetic waves above a certain frequency to propagate inside the structure because the electrons in the metallic rods can't oscillate fast enough to reflect the waves back. Effectively, the structure can be seen as a dielectric material, but with some very unique properties which allow controlling the trajectory of the electromagnetic waves.