It seems as if data transmission wasn't FTL.
Initially, let's assume it wasn't FTL. In this case, it's easy to see the theory of the maneuver. During its warp jump, the after-image of the Stargazer in its original position appeared absolutely real. Coming out of warp, the Stargazer appeared to be a second, different ship. The Ferengi computers had been set to target the first ship, which still appeared to be there, so the computers had no reason to change that setting; when the Ferengi fired, their shots went for the wrong ship. It would also have caused confusion and alarm on the Ferengi bridge to see a second ship arrive. For both these reasons, it was just enough time for the Stargazer to attack before being attacked.
(I believe there would have also been a third Stargazer image, moving away backwards, which was the after-image of the ship in motion as it arrived. It seems like this could be used to detect the maneuver even using light-speed sensors; if you use the backwards-moving image to understand the trajectory of the ship that just arrived, and it coincides with the position of the first ship, then you have good evidence that the two ships are the same. However, the Stargazer could have prevented the after-image being immediately useful by taking a curved path towards the Ferengi.)
The screenshots in the question are from 1x09: The Battle, where the Stargazer attempts the maneuver against the Enterprise. They are not an accurate depiction of the maneuver. For one thing, the distance between the ships is much too low. Light is really fast -- you'd need 23.5 Earth widths between the ships to get even 1 second of light-speed delay! We can dismiss the depicted distance as artistic license because if the ship had started at a proper distance it would be too tiny to see on screen. However, another flaw is that the ship on screen can be seen starting to move, seen moving forwards, and then seen stopping, which does not really make sense. For an in-universe explanation of the on-screen images, we can imagine that since Picard was under mind-control, disoriented, and operating the ship without a crew, he bungled the maneuver.
Anyway, the point in that scene is that Riker and Data were worried by the prospect of the maneuver being used against the Enterprise and Data states there is no defense against it.
However, the Enterprise clearly has FTL sensors, at least long-range ones. Long-range sensors do not seem to be an obscure technology, so the Ferengi should have had them too, so how could the maneuver have ever worked? Note that there are a variety of different sensor technologies in the Star Trek universe, many light-speed and some FTL. We can hypothesize that the FTL sensing techniques are simply too imprecise to be useful for targeting ships in close-range tactical battles and thus are relegated to long-range use. Often we see that battle damage causes a ship's short-range or long-range sensors to go offline independently from one another, suggesting that the two technologies do not substantially overlap in their design.
There is a simpler explanation given in the Star Trek Fact Files, which was a Paramount-licensed magazine series of encyclopedic information. In an issue from 1997 it has an interesting analysis of the Picard Maneuver. It implies that the Ferengi ship simply did not have FTL sensors at all, which it describes as a "serious deficiency". It also states:
The tactic is only effective against opponents who rely solely on light-speed sensors, and offers no advantage against opponents with faster-than-light detection technology.
It doesn't explain why Riker and Data were worried about it being used against the Enterprise.
I've reproduced the two pages of the article below. (Regarding copyright, I believe that posting two relevant pages for discussion out of seven thousand pages in that whole series, particularly when it is out of print, is minor enough to qualify as fair-use, although I'll remove them if asked.)