Interstellar transmissions are low-bandwidth, and they degrade easily.
The Death Star is an enormous engineering project. Its blueprints fit into a physically small volume, but so does C-3PO's vast linguistics database; the amount of data in the Death Star plans must be astonishing. The Rebels hope to find a way to exploit Galen Erso's sabotage, but they don't know where in the plans to look for vulnerabilities,(1) so they need to send the entire set.
With Imperial forces in hot pursuit, the rebels didn't have time to transmit the data with enough redundancy to protect against signal degradation and ensure the plans' reception in usable form. For example, asteroids interfere with interstellar transmissions:(2)
PIETT: The Emperor commands you [to] make contact with him.
VADER: Move the ship out of the asteroid field so that we can send a clear transmission.
— The Empire Strikes Back
These lines of dialogue begin at 0:24 in this clip, followed by a flickering, jittery, grainy projection from the most powerful person in the Empire:
Since the Emperor — with all of his resources — can't send a steady, detailed faster-than-light transmission, the rebels must have faced even greater limitations.
Vader's conference with the low-resolution, flickering images of the ship captains at the start of the clip takes place in realtime, with no speed-of-light lag. Although at least some of the ships are close enough together for lightspeed communication, the simplest explanation is that they are using the same faster-than-light communication system here that they would use at interstellar distances.
A similar effect appears about 79 minutes into The Phantom Menace. While on Tatooine, Padme uses her royal yacht's holographic communicator to play a message from Sio Bibble on Naboo. Bibble's low-resolution image constantly wavers, and his voice is distorted. Even the most stable frames in his message are blocky:
Darth Sidious' holograms in The Phantom Menace show the same characteristic jitter and pixellation. He was concealing his identity throughout this film, though, so the poor image quality in this case may be due to more than technological limitations:
(1) The attack plan that the Rebels hurriedly devised in Episode IV depended on Galen Erso's sabotage and on two design flaws (the shield permeability and the exhaust port), and ultimately it only succeeded because of space wizards. The Rebels may have been able to devise a safer and more reliable attack strategy if they'd had more time to study the Death Star plans.
(2) The fleet's use of weapons may be a contributing factor, but if it was the only concern then the ship would not have had to leave the asteroid field.