I understand Blizzard's statements, in the simplest terms, to mean that early in Mists, healers will have smaller health pools to refill from damage in smaller increments or over longer periods of time. In Mists' last tier of content, larger health pools will be reduced by damage more severely, more frequently and/or for prolonged periods, but appropriate gear with Spirit comparable to Intellect will provide the regeneration necessary for a healer to perform.
(I suppose it's possible that larger casts may cost slightly more while regeneration compensates, so that the model remains stable from beginning to end — e.g., high Intellect draws 20% more for a Greater Heal, but high Spirit replenishes mana so quickly that a player is simply aware that he can cast a big enough spell to keep up with content).
This suggests that mana regeneration will be much faster than players are accustomed to, although I don't see an explicit acknowledgment of that from GC's posts. Proof of concept is also vague. My guess is that reasonable players will accept the system if they have more details, even if they can't articulate which ones will satisfy them.
As for why, I think I understand: Intellect is such a manifold stat that it dominates the decisions of developer and player alike. The analogy might be "gain staging," which is balancing devices in a sound signal chain to prevent distortion (if too strong) or noise (if too weak). Right now, Intellect is so overpowering that Spirit doesn't matter much for most specs, while it's disadvantageously important for a couple — and GC envisions that "[f]or [Mists] healers, there should not always be a clear cut answer."