You have a couple of polar opposites at work here, making this sentence a definite paradox. You cannot have silence if a voice is present. That much is evident. However, if you look further, you will see that the author says that silence fell on the room like a clap, also something that makes noise. Whether it's the clapping of hands or boards, a clap is definitely a noisemaker, and noise prevents silence.

Teachers will tell you that almost everyone has had an experience where they have said something extreme or shocking, and the immediate result is silence from everyone else in the room where the comment took place. It sounds as if this is the context in which this quote is delivered -- someone just said something that made everything else fall utterly silent. The way the quote is presented makes it paradoxical, however.

Carrying something or someone keeps it from falling or touching the ground (only one set of footprints in the sand if you are being carried--your feet don't touch).  So, you've got two ideas that are really at odds with one another...the idea of a carried voice (either silent or the notes are held out for a long period of time) and silence falling quickly like a clap (the voice was dropped?); and the second idea of sound vs. silence simultaneously.

I think you can interpret this quote in at least two ways:  If you believe the voice is being carried/cradled carefully so that it will not fall or make noise, then the quote means that the voice was carried, thus hushed, and therefore silence fell quickly on the room like a response to a clap for attention.  If you believe the voice is sounding out like a singer holding a note (the voice carried over the airwaves), then the silence fell equally as quickly because of the voice's command for attention--perhaps it is a beautiful voice and everyone is mesmerized in the time it takes to clap.

Either way, the two contradicting ideas make this a memorable quote.  It is an effective trick.