It's not offensive, it's just not right for business. Outside of work, men use ore (tough and manly) or boku (boyish and charming) in regular conversation. Some my co-workers even use ore at the office if they're making a joke or something.

What would be the correct pronoun for a business setting? My knowledge of the language is rudimentary at best (several semesters in High School); we only really learned watashi/atashi/boku (and were reprimanded if we used the wrong pronoun for our own gender).

In doubt use watashi, it's hard to go wrong with it. The more formal watakushi is also good but you need the keigo to match, and there is such a thing as excessive politeness, even in Japanese.

What does the use of 'ore' imply and what makes it rude? Is it a snobby way of referring to yourself?

I've never heard the word before this thread but I'm inclined to think it's a bit like "lads" in British English.

It's really... boastful. Cocky. Self-important. He's using it in sentences where he could just as easily get away without using any pronoun at all.

The closest thing I could think would be to conduct a business meeting in which you only describe your actions and refer to yourself in the third person, and then only as "Big Mark".

"Big Mark understands your position but thinks you're being a bit rude about all of this. Big Mark thinks we should cooperate more" etc, etc

It's not snobbish, it's cocky/boastful. "I'm hot shit" sort of way.

Why is it that way? I don't know the history, but the short answer is, because that's the way it is used.