what, why, how on earth

I don't think it's offensive, you use it when you want to emphasise something..at least that's what I was told

I would be a little cautious that it was only used with people I knew quite well. It could be interpreted as a criticism of someone's statement(s) or thoughts.

Yes, I agree - it can be criticism. It can also be admiration. It's used to express being very surprised by something.

"How on earth did you reach that conclusion?" -- This sounds like you think the person is stupid for thinking this.

"How on earth did you get that done so quickly?" -- This sounds like you're amazed/impressed at the person's accomplishment.

So... as Feeb

Thanks Pucca, Feebs and GG for your valuable comments and suggestions.