What does it mean for an electromagnetic structure to be resonant? There are many electromagnetic structures used in microwave engineering and EM devices. For example, patch antennas, metamaterials made from unit cells, etc.
When they design structures like patch antennas, split-ring resonators in metamaterials, and many other similar microwave structures, they want the device to resonate at a required frequency, and somehow, the operation of the device depends on it being resonant at a given frequency. 
What is the meaning of this resonance exactly? Why, for example a patch antenna or a split-ring resonator needs to resonate to function properly?
 A: In the context you cite, a resonant structure is one that sets up multiple reflexions, generally in a cavity formed between two or more reflectors, such that the power of the reflected wave is a high fraction of the incident wave. 
What this means is that highly restrictive electromagnetic boundary conditions are set up, which means that only waves near to certain discrete eigenfrequencies of the structure can be significant within the structure. At these frequencies the multiply bouncing waves inside the structure interfere constructively with one another. At all other frequencies, the mutual interference is destructive, and the EM field is efficiently excluded from the structure.
The prototypical resonator you should study to build intuition for these ideas is a resonant cavity between two partially reflecting mirrors. You can study such a resonator by looking at my answer here, in the section called "How Do Mirrors Shape a Spectrum?". I discuss the characterisation of a partically reflecting, lossless mirror here: this should be useful to your study, and I use this characterisation to study another simple resonant device, the optical ring resonator here.
