Why is Eobard Thawne in Legends of Tomorrow?

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.

Spoilers for Season 3 Follow

Thunderforge is right about him being the reverse flash from flashpoint since it has been revealed to us in legends of tomorrow. However i do believe that writers' of "The Flash" logic is broken. In the end of season 3 episode 1 we see that the reverse flash from flashpoint goes back in time and kills Nora Allen (the previous flash from season 2 and the original reverse flash he is trying to stop from killing his mother disappear) but then returns to Barry's present leaves him there and then runs off back to the future or wherever he wanted to go. Since he did that, there is no Reverse flash in the past (The original reverse flash had disappeared) after Nora Allen dies (Unless the flashpoint reverse flash returned back to the time nora allen died and impersonated wells and got erased from existence (in which case means that he couldn't be on Legends) which he didn't need to do cause he had his powers and could return to his future) who kills the real wells and impersonates him and thus Barry shouldn't have become the flash.

What the reverse flash from flashpoint should have done was stop the Barry from season 2 from stopping the original reverse flash killing barry's mother and therefore the original reverse flash would go on to impersonate wells and create the flash. This would leave the flashpoint Reverse flash (RF)as a time remnant and then it would make sense that the flashpoint RF is on the Legends Of Tomorrow trying to rewrite reality with the spear of destiny and create Eddie Thawne and therefore not be a time remnant anymore being hunted by the black flash or time wraiths.