Helium makes your voice high because its lighter than air. If you breathed a gas that was heavier than air, then spoke, your voice would sound lower than normal. (Unless the gas was toxic, in which case you might be so busy choking to death that you wouldn't have time to speak. Don't try this unless you know the stuff you're breathing is safe!)
Why should the lightness of helium or air matter? Because the speed of sound depends on the weight of the helium or air or whatever the sound is traveling through. Sound travels faster through helium than through air, because helium is lighter than air. The sound of your voice is caused by the vibration of vocal chords in your larynx (or whatever they call it! I forget the name), which makes the air in your larynx vibrate, thus producing sound. How fast the air vibrates is determined partly by the speed of sound in air, so that if the speed is higher, the vibration is faster. So helium, with its higher speed of sound, makes your voice higher.
A related effect is well known to anyone who plays a stringed instrument. The high-pitched strings are much narrower than the low-pitched strings. This is to make the high-pitched strings lighter, and the low-pitched strings heavier. Just as the speed of sound depends on the weight of the air, so does the speed of vibration of the string
depend on the weight of the string.
Heres a little experiment you can try, if you happen to play a wind instrument: Get yourself a helium balloon, and have your instrument ready. Play a note the way you regularly would. Then take a breath of helium and try to play the note again. What should happen? (I once tried this with my trumpet. Its good fun!)