Probably the same way it worked when games were split across discs: lots of data and assets repeated across discs.
That's a bit different though. Once you got to the end of a disc, you could just pop in the next one and keep playing.
With episodic releases, you reach the end of an episode, and then what? Do they just block off an area and say, "Now you're in free roam mode! Feel free to go back and do side quests that you missed!" and then exhaust all that content and wait another year until the next episode comes so you can chase Sephiroth a little bit more before the next road block? And what about when the story originally took you to older locations to advance the story. Do you block off those locations during "free roam" until the next episode is out?
All the episodic games I've played just end the game at the end of an episode and the only way to go back would be to reload an old save or do a chapter select. I don't think that would work in an RPG.
Making a big RPG like FF7 episodic seems like a really weird thing to do. They're going to have to frame the episodes really carefully. What I fear most is that they'll just make the game super linear (more linear than the game originally was) to solve that problem.
I'm disappointed with this "multi-part series" news, but I also am curious to see how they even pull it off. So many questions in the air.