We haven't made a video, but I think I can figure out what you're asking anyway.
You're probably picturing things in a special relativistic coordinate frame. Let me play along. Light that's reaching us now must have left its source at time t-d/c, where t is the current time since the big bang, d is the distance that the source was when the light left it, and c is the speed of light. We know t=13.7 bil years. Now if everything we can see started out essentially at the same place at time zero (the bang) and nothing travels faster than light, the farthest that source could have been from us is c( t-d/c)=ct-d. If that distance is d, we'd get d=ct/2 and the time of the light started would only be t/2. That's not 13.7 bil years ago but only 6.85 bya. So how can we say that the light is almost 13.7 bil yr old? (at least I think that's what you're wondering.)
Staying in this same sort of frame, there's a bit of a hitch. What we really want to know is not how old the source was in our coordinate frame but rather how old the source seemed to itself. Now this region was traveling away from us at almost the speed of light, so special relativity says that its clocks seem radically slowed-down to us. Its age according to its own internal processes is still much closer to zero than to 13.7 bil yrs.
A more serious calculation would use general relativity, which allows a wide variety of different space-time coordinate assignments. The most standard, however, is the co-moving coordinate frame, in which time in each region is kept by a clock at rest with respect to the broad average of all the stuff near there. With this choice, the time at the start of the background radiation is still only a few hundred thousand years, not billions. So it starts very young and ends up at t=13.7 by, so we can say it has traveled for almost 13.7 bil yrs. I believe that in this coordinate system the explanation for how it took so long to get here rests on the expansion of space.
(published on 06/26/12)