The video clip is slightly misleading, and presents a bit of a simplification of what would've actually happened. In reality, what the MRO would've received was a carrier signal, which would then be transmitted back to Earth and analyzed by scientists and engineers for clues about the success of the mission.
As explained by Michel Denis, Mars Express Spacecraft Operations Manager, and reported on the ESA website:
"Only Odyssey can receive, decode and then relay to Earth the actual telemetry data coded into the those signals. Conversely, MRO and MEX will save on board 'open-loop' recordings," says Denis.
Michael Khan, a mission analyst at ESOC, confirmed this for Mars Express, as reported on the Mars Express blog:
"All that Mars Express will receive is the UHF [radio] carrier signal. From apparent changes in the received frequency, one can deduce at which speed relative to the MEX the landing craft is moving. Or -- hopefully this will not be the case -- if suddenly the signal stops, then we have an indication of when things went awry," he points out.
"Mars Express can and does relay data for surface assets. But that is long after landing and requires MEX to pass very low over the landing site; not the situation we will have on Monday, 6 August."
And they went into a bit more detail into the open-loop recording process in a follow-up post, "What is Open Loop Recording?":
In open loop recording, we don't try to decode the bits and bytes being sent by the descending lander but instead try and listen to as much of the radio spectrum as we can, hopefully detecting the tone of the lander's transmissions within this spectrum. Think of it like listening to a crowd of people – you can either focus on the words one person is saying, or listen to the whole crowd to get a full picture of what's going on; that's what we'll do with open loop recording.
On Mars Express we'll use our UHF Melacom radio to listen in on the UHF part of the spectrum – usually used on Earth for radio and television transmissions; it's also used at Mars as the frequency that different orbiters and landers use to talk to each other.
By using this technique, we will see the trace of Curiosity's signal in the UHF and X-Band ranges and watch as it drifts and changes in response to Doppler, in this case not just because Mars Express is moving overhead but also because Curiosity will be changing velocity as it slows itself to a soft landing on Mars. This will give us a great insight into the progress and success of the landing.
This process of analyzing the signal is what would've taken hours. Of course, it wasn't needed as Curiosity was able to transmit data to the Mars Odyssey orbiter, and it likely won't be needed again for the rest of the mission.