From the point of view of speech, the vocal cords can be conceptualised as two membranes that vibrate upwards and downwards to make sounds.
When the vocal cords move up and down, they encounter resistance in the form of the air around them. It's a slightly less dramatic phenomenon, but identical in principle, to trying to stir liquid toffee up and down in a pot - it's hard work!
When we breathe in helium, some of it replaces the air that we had in our lungs moments earlier. Now helium is the second lightest element (its atomic weight is 4), and is considerably lighter than most of the atoms the constitute air, mainly nitrogen (atomic weight 14) and oxygen (atomic weight 16). Therefore, standard-issue air has a much greater inertia than helium gas does. It's a bit like replacing the toffee in our pot with water - the job becomes much easier.
Now, would would happen if the force trying to move the vocal cords stayed the same (how would the respiratory system know we've just inhaled helium?), but the resistance to their movement decreased? Why, the vocal cords would move up, and down, more quickly.
The quicker the vocal cords vibrate, the higher the frequency (frequency is defined as the number of waves per second) and so the higher the pitch of the person's voice.
One word of warning - don't breathe in helium for any length of time; if you really want to play around, just a breath or two should be fine. As I said, the helium displaces the other gases that make up the air - including oxygen. And life without oxygen isn't possible for us humans!