I understood nothing except that the king and giuseppe are snake-like bastards.
Let's put simply what I understood:
The Twin Kingdoms are ruled conjointly by two royal families, but the Unification faction, which includes both the King and prince Giuseppe wants to make it ruled by only one. Enter Francesco which has the lineage, and magics, of both; making him the obvious choice.
But that is a can of worms on its own. Many nobles rightfully will oppose such a move, so they plan to divert their attention with Zenkichi, Zenjirou's an Aura's son, which also has the blood of both the Capua and Twin Kingdoms bloodlines, making him a potential holder of at least two bloodline magics, one of them their own. This is a problem by itself and while the nobles are occupied with him, the Unification faction will make their move.
Then we have the dukedoms, which is a headache on their own. Simply put, they're divided into the nomad dukedoms which are poorer and more military-oriented, and the "aristocratic" dukedoms, which are richer, closer to the royals and more populous. They don't fully get along, and how to deal with them before the situation blows up drives a wedge between the king and Giuseppe, who favor leaving the nomad dukedoms to rot and degrade their rank, and Largo, which proposes aiding them for them to keep their rank because they're very useful.
And then we have Zenjirou who, like usual, is roped into this mess without wishing to. Sometimes sucks to be him.