I couldn't believe that the doctor said that helium was being used as an anesthetic either and as soon as I heard that I rewound it. I don't think that's what he says, but since they cut to him in the middle of a sentence it's hard to figure out exactly what the first couple of words are.
Anybody who'd ever inhaled helium could tell you you don't get high off of it though, and it doesn't have any kind of anesthesia effect.
According to the article on nitrogen I linked to the way a gas like nitrogen or helium would kill you is through asphyxiation, however you wouldn't be aware of what's happening until it was too late.
When your're breathing stale air or holding your breath you get the urge to breathe more not because of a lack of oxygen but because of a build up of CO2 in your bloodstream. Since nitrogen and helium are odorless and tasteless you would have no idea you were breathing it and wouldn't know you were suffocating because your body is still excreting CO2. No C02 buildup, no alert to suffocation.
Then, by the time you know you're feeling fucked up it may already be too late. The article cites a NASA incident where an inspector walked into a fuel tank unaware the oxygen had been purged with pure nitrogen. He was all walking along then bam! He hit the ground. Someone else saw this and went in to help him unaware too of the nitrogen situation, then bam! He passed out too.
By the time anyone figured out what had happened they were both dead.