What happened to the second Barry in Rogue Time?

This event seems to be the singular instance of this happening in all the time travel involved in the Flash, so it's likely that this was just creative licence, because trying to deal with a second Barry in this episode would have been too complicated on the writers' part.

However, if you want an in-universe explanation, I would say this:

The "other" Barry is from a future where Barry went back in time, but because our Barry saw him, it led him on a path where he ended up going back in time (where otherwise he wouldn't have done) to create a past where he didn't go back in time, causing the other Barry to be erased. When he later became the "other Barry" the positions were reversed and the "original" Barry disappeared because he was from an aborted timeline. To elaborate on this, when Future Barry went back in time and was seen by his earlier self, he disappeared because he was transported immediately to HIS version of events that occurred as a result of the original Barry going back in time. To put it a simpler way, because Barry's time travelling was a predetermined event from the beginning, the events we see from the moment near the beginning of Out of Time and the moment he leaps at the end, are THEMSELVES the alternate events that only Barry will ever have experienced. In a sense, during this episode, Barry is truly "Out of Time".

This is probably an example of the parallel/alternate timeline.

When Barry time travels at the end of "Out of time", he gets transported to an alternate timeline. Think of this analogous to you loading a previous checkpoint in a game.

This way, his time traveling spawns a new timeline, where the events are simply reset to that point.

This is why we do not see 2 Barrys.

As for the original timeline, think of "Back to the future". When Marty lands in the dystopic timeline...

Now just imagine the same scenario, with Flash in it.

Due to lack of answers from the show, and contradictory time travel effects, or effects that vary based on method of time travel (Running Fast vs Intentionally creating a singularity and then just navigating it), it may be possible that Barry never actually time traveled in Out of/Rogue Time.

Ignoring that we know the Flash can travel in time in multiple canons (through various means), Both what Barry experienced (AND what Cisco experienced, his meta powers being close to clairvoyant in nature), can be attributed to a Premonition. In that single moment where he saw himself during Out of Time, he was confused and disoriented. It could be that he glimpsed a possible future, which is what we see in the episode, like a dream, a precognitive vision. It never actually happened, and Barry never actually travels through time. As far as the story goes, this still has the same effect. Think Final Destination instead of Back to the Future.

Alternatively a way to explain how only one Barry exists in Rogue Time could be a X-Men Days of Future Past, TNG: Cause and Effect, Interstellar type situation. Barry never physically travels back in time, he just sends information, a message, his consciousness, soul, etc. This either creates two time lines, one where Barry-1 still struggles to save Central City from the Tidal Wave and another where Barry-2 averts it, or it creates a paradox which wipes the first timeline out of existence. In both cases, the changes take place because of a sense/feeling Barry received instead of actual time travel.

Both of these scenarios are quite common in both comics (Flash included) and live action film/tv plots, and can easily be the reasons for what we see on screen.