This is an earlier time point Eobard. Sort of a Flashpoint version who found out what was going to happen to him in the future. All of the parallel time lines still exist for a period of time after the past being changed eliminates them. That is why on Legends we get changes that gradually phase into existence like Stein's daughter or phase out of existence like Ray and Nate losing their previous knowledge and abilities due to Star Wars not happening. When faced with being wiped from the timeline Eobard fled to a different timeline since speedsters are able to do that without a time ship. Speedsters interact with time change differently than mere mortals due to being tied into the eternal speed force. Normal propagation of the time change isn't able to wipe him out so the speedforce itself is using its enforcers to correct the aberration.
And the reason that the future Eobard vanished the moment Eddie shot himself is because at that moment there was an open portal to his own time that effectively linked those two time periods so that the timeline change was instantly able to affect the future and didn't have to travel there as a time wave. It wasn't a propagation effect but rather an instantaneous change.
I swear I could make a career of explaining ways around the writer's inconsistencies.
I get that this RF being a time ghost or whatever from Flashpoint makes sense. Barry pulled him from time, changed it and then brought him back to a timeline he never existed in. If that was the only thing - I would have no problem.
My 'thing' is the original event where Thawne's ancestor was removed from history - everything that he did should never have happened in the first place if he was never born.
He never would be born to come back and kill Barry's mom, the original Wells should have been alive and well, and no one but Barry should have remembered anything that happened with RF.
But everyone still remembers RF, the fake Wells, the real Wells is dead, etc. He even left a video confessing. That just made me scratch my head.
Now I will give them a break as they had that massive wormhole that tore the fabric of space-time at the end, so at least Eddies death caused something cosmic to happen.
But still my favorite show on right now.