Eobard Thawne explains his existence in Legends Season 2 Episode 10 "The Legion of Doom"
Spoiler-free plot-point from that episode.
Thawne: My ancestor [Eddie Thawne] killed himself in an effort to erase my very existence.
Darhk: Well, you look pretty spry despite it.
Thawne: The Flash pulled me from the timeline. He held me captive for months. Then when I finally got loose…
This matches what happened with the Eobard Thawne from The Flash Season 3 Episode 1 "Flashpoint". Barry Allen created an alternate reality by removing Eobard Thawne from the timeline at the key moment when Thawne attempted to kill his parents in the year 2000. Ultimately, Barry had to let them die and restore the timeline, and this Eobard Thawne ran off.
Note that this is different from the time renmant of Eobard Thawne seen in The Flash 2x11 "The Reverse Flash Returns". The short answer is that a time remnant is a replay of a speedster's previous time travel, despite being erased from existence. This Thawne is a completely new creation with a new future ahead of him.
But as we find out in this episode, being the only remaining survivor from a timeline that no longer exists has consequences.