The non-video explanation is, Pokemon bank connects to the eShop twice per use of Transporter or Bank, to verify your NNID and account status. Kinda inefficient, and when the release hit, it obviously was a mass scramble to move yo mons. The whole thing came to a head when midnight EST hit and some kiddos got a present early, or the parents were trying to set up the 3DS or Wii U for the morning. That meant there was only bout a 6 hour window for anyone to "buy" the Bank app before all hell broke loose. I was one of those lucky sonofabitches.
Seeing as bank and transporter now both run very smoothly for those who got it, I am now expecting to soon see a patch for the program, to make it even smoother, and eventually a staggered release. If I was a bigwig at Nintendo, I would limit the number of "purchases" to 1000 per day per Region, to offset the server strain. Also buy a fuckton more bandwidth.