"Hare Ram Ram" literally means "Pray the name of Ram several times", where Ram is probably an ancient prophet in Hinduism. It is a prayer often used for people who are about to die. When Kelso's stick got into Bakshi's butt, it was such an unusual experience for him that he thought that he was going to die.

You can clearly hear that he is saying "Hari Rum Rum", however that doesn't mean anything in any Indian language. I always assumed that it was either an ad-lib by Sellers or it was thrown into the script intentionally as a nonsensical term, as it does sound very Hindi-esque (Hari Ram is a common Indian name). He's using it as an equivalent for an "Oh god" as is evidenced by him putting his hands together as he's saying it.
It's comparable to how Mike Myers used the phony Indian greeting "Mariskahargitay" in The Love Guru. It definitely sounds super authentic, unless you've heard of the Hungarian-American actress named Mariska Hargitay.

I agree with the other poster who claimed he's saying "hari rama rama". Sometimes Hindus don't pronounce the last A in Rama and they pronounce As different then westerners, so it sounds like "Harry Rum". Remember, he's also looking to the heavens up as he's saying this.
But I'm not sure why getting anally probed would cause him to pray to God.

BERNIE NUM NUMS!!11!1!2@22@!!1!

the bird thing says birdie num nums, when he gets the stick up his ass he goes hari rum rum!
i have no idea what it means but i say it to my mum all the time haha!

He acutally says Hari Rum Rum when wyoming bil kelso pokes him in the but with the pool stick!

The seed container from which Hrundi is feeding the bird is actually labelled "birdy num num" which is where he gets the saying from.

When he's feeding the bird he says "birdy num num" which I think is where you are getting confused..

He actually says Hari, Rama, Rama. Which is sort of like a prayer, like, oh my god.