He He He(lium)

So why does inhaling helium make one's voice higher?

Actually, it doesn't.  Nope, helium does not cause you to talk at a higher pitch...what it does, precisely, is elicit a squeakier timbre. 

And here we get into some acoustics...hmm if I were being paid for this, no doubt I would do a bit more research, but off the top of my head...so timbre, or tone quality, is basically anything that isn't (1) pitch or (2) loudness.  It's what makes a saxophone different from a violin, even if they're playing the same note.  And I belieeeve it's got something to do with fundemental frequencies and harmonics.

In addition to distinguishing jazz and classical, timbre also helps us differentiate voices with and without good old periodic element He.  You may have noticed that when helium's in the vocal tract, basically the only sounds that are actually affected are the vowels.  Well that's not quite true - any voiced phoneme will do - but the point is, you can hiss like a snake as long as you like, and you won't sound any squeakier until you vibrate those vocal folds.  That's because of how helium works.  And finally I'm getting to my answer. 

Inhaling brings air to your lungs (of course).  Speaking then releases this gas out of your lungs in specific, controlled streams and bursts.  That's all well and good, but keep in mind that helium is lighter than air, as any first-grader who's had a birthday party can tell you.  This lighter, less dense medium plays a trick on the vocal folds - and on the brain that's hearing the sounds these vocal folds are making.  When we hear speech, we are hearing the combination of two things - the vibration of larynx, etc. and the vibration of air molecules.  While helium doesn't alter the former, it does change the latter - the speed of sound in helium is significantly greater than that in a denser material.  And it is this property that alters the resonance patterns - the timbre - of what we hear.

For more info (and some amusing Australian? sound-bytes...especially the last two), click here.

*NOTE: Inhaling helium, almost by definition, means not inhaling oxygen...so don't overdo it, okay?  Also, definitely don't try inhaling anything ever directly from a pressurized canister or container.  And furthermore..! If you're especially brainy (or at least, if I've explained this with a modicum of coherency), it would probably occur to you that inhaling a high density Noble gas - say, Xenon - would decrease the speed of sound and result in a deep, booming baritone (or something of the sort - I can't figure out if singing voice is a factor of pitch or timbre).  And you would be right...however, keep in mind those floaty helium balloons, and consider how, exactly, one might empty their lungs of denser-than-air Xenon once it has been inhaled.  We're not talking carbon monoxide here, but...play it safe, okay kiddos?