At the end of Star Trek Beyond, how did Uhura know who Krall was?

Lt. Nyota Uhura is a communications specialist, with a talent for languages by ear. This extended to her analysis of Krall, as she is the only one he speaks to at length aside from Kirk. Specifically:

The Federation has pushed the frontier for centuries.
But now enough.
This is where it begins, Lieutenant.
This is where the frontier, pushes back.

And in the video log when Uhura analysis it, there is a bit where Edison, back towards the camera, says:

...expand the frontier...
rewind
...expand the frontier...
rewind, slow down
...expand the frontier...

Nyota plays back the video multiple times, slowing it down.

She also sees first hand Krall absorbing the life forces of two human crew members, and his skin and voice changing accordingly. Both become less bumpy, grungy, more smooth, more human.

Between changing the pitch of the Video, and Krall's changing voice, and the specific keyword of Frontier, it clicked in her head.

She still didn't know exactly how or why, but she knew Krall is Edison from watching the video. Right after the rewinds:

Uhura: It's him. Scotty, I need you to link to Franklin database. Find out what you can about Balthazar Edison.
Scott: What?! The Captain of the Franklin?
Uhura: Yes.
Scott: He is long dead.
Uhura: No he is not. I do not know how, but Edison is Krall.

Actually, I thought she recognized his voice. (remember - great communications officer, knows three dialects of Romulan, ...)

As shown in the film, as Krall absorbs life force from others his appearance changes, most notably his skin appears to become smoother, and later in the film he was able to absorb enough life force to almost completely regain his human appearance. Having conversed with Krall in person and seeing him use his abilities, it's possible that even a glance at the rescue footage the USS Franklin had in its databanks was enough for her to postulate that maybe Captain Edison may be Krall.

After the idea comes to her and Kirk upon closer examination of the video they initially found, they find Edison's logs and learn of his ultimate fate. This could be corroborated with the fact that he seems to have very intimate knowledge of the Federation itself, and even mentions to Uhura that his ultimate goal is to destroy the Federation.

I watched the film a little more than a week ago, and I've hardly an eidetic memory, but I got the strong impression that it was a specific phrase that Krall had told Uhura, which she then overheard in the background of a video. I cannot recall the phrase itself, but I'm pretty sure it was hearing this phrase and isolating its source that led to identifying Krall as Edison.