You'd think that the absence of a bass line for almost half the song will result in a boring first half, but it doesn't. In fact, I'd go so far as to say that introducing a bass line that early into the song will destroy it. The guitar and piano/synth lines that accompany the melody until the end of the first chorus allow us to hear the melody, which is what steers this part of the song. There's also a drum line -- that thin, repetitive beat that sounds like bongo drums during the verses, and the tambourine during the chorus. It gives the song some body, and it's enough to act as a foundation. The chorus is quiet but urgent, catchy but not in-your-face. The layering of several, thin, vocal tracks also plays a big role -- the intensity of the chorus itself is in the harmonies. You hear all of this, the catchy chorus, the harmonies, because of the bare instrumentation. The melody doesn't have to compete with dozens of other, potentially more obvious, sounds, so it doesn't have to scream to make a point. The first part is a very minimalist approach to the instrumentation (as far as pop songs are concerned), but it works. "Hush" uses elements sparingly, yes, it gives space to breathe, but between those spaces are all sorts of kicks and punches.
"Hush" works precisely because of its two parts, because it is just as much a natural progression as it is a juxtaposition between the subdued first part and chaotic second. It's a song that, in keeping with its title, is quiet in all the right places. But at the same time it knows exactly where, when, and how, to pack a punch.