Why is it rude to tell other people that they're being rude?

In a respectful manner, it's not. If it's not respectful, i.e. 'Hey, stop being a dick' implies they were being a dick, insulting them at the same time insulting you. If you say 'Would you kindly not say insulting terms' or something (no one would ACTUALLY say that, or maybe you would) then it's not rude. To my knowledge, anyway.

If someone is being rude, you should take them to one side and tactfully tell them that they are being rude, and explain to them what they did wrong.

If you just pull them up in front of a lot of people, it can cause them embarrassment, and it can cause anyone who you are in the company of embarrassment, and it can make you look like you are being confrontational.

I don't either. It is perfectly alright to point out to someone that they are being rude, if they are! But most people are either scared or embarrassed to deal with such people so to avoid a scene.

Sounds like one of those " damned if you do & damned if you don't " scenarios.

If you can't tell them them they're being rude, how will they know?

Someone has to tell them, may as well be you.

It doesn't matter if it is rude and you don't necessarily have to be respectful depending on the circumstance.