Thawne is a time remnant. If Eddie erases Thawne, he's never born to motivate Eddie to shoot himself in the first place. It's a temporal paradox. The Speed Force fixed this by keeping him alive up to the point of his death by Eddie so the timeline remains intact. Therefore when Eobard accidentally shows up in Season 2 before murdering Barry's mom, he hasn't yet met the team and everything.
The explanation back then was time remnants. In order for Season 1 to happen, Eobard needed to exist too. So despite being erased from existence, a remnant of Eobard was allowed to exist by the Speed Force to ensure Season 1’s possibility.
Season 2 reverse flash had never previously been to the 21st century, the reverse flash from seasons 3 onward exist because of flashpoint (although it was supposedly rectified in the 2nd season finally of legends of tomorrow, so it is not clear how he got to earth X
@Nsawyer95 After Sara used the Spear of Destiny to free the Black Flash, Eobard was subsequently erased from existence by the latter and absorbed into the Speed Force as during the scene you can see yellow-orange particulates. I believed that after Barry escaped the Speed Force with the help of Team Flash, Thawne also escaped and used the Negative Speed Force to travel to Earth-X and kill that Earth’s version of Harrison Wells and transform into him like he did the Earth-1 Harrison Wells, thus leading into the events of Crisis on Earth-X.