Feist's version of "Tout Doucement" is, for the most part, a basic I-VI-II-V chord change sequence. This is very common in pop tunes (in fact, the Muppet's "Mahna Mahna" is the same set of changes). The song is in B flat, making these changes Bbmaj7-Gmin7-C7-F7. The chorus also ends on F7, but instead of going back to B flat, it moves up to F# (the new V, essentially) and finally lands on Bmaj, the new key. Her voice also goes from playful to impassioned, which only adds to the intensity that the key change offers.
The song was written by Emile Jean Mercadier and Rene Albert Clausier.