Is answering a question with "Why, " rude?

In a book I've read some characters are answering questions with "Why, ", e.g.

Why not come to me? you ask, and I must answer, Why, because I did not trust you, my lord.

and they never mean it seriously, always with some insolence. However, the amount of insolence and how rude it really sounds I cannot grasp.

Can I use such a construction in an informal speech without any offence meant? In a not-so-formal letter to a boss, for example?

I hear "Why" used as a mild intensifier. It has some overlap with other opening phrases such as "You see, ". In some contexts, it could be condescending.

Here's several different examples of what "why" adds to the statement.

A: Who is that?
B: Why, it's my long-lost brother! (Surprise!)

A: Could you fetch me a drink?
B: Why, of course. (Whimsy. The speaker is pleased to do so.)

A: Daddy, will I be as tall as you when I grow up?
B: Why, you'll be even taller! (Emphasis! Especially if "talking down" to a child)

It is easy to sound archaic or condescending, so take care where you use it.

Beginning an answer with the word "why" is not inherently rude; the answer could be rude because of its content, of course.

This construct is definitely somewhat archaic, and I've seen it more often in British contexts than American.

The second why in the quote is an interjection, an expression of mild surprise, or as another answer notes, a mild intensifier. In office communications, informal or not, interjections probably are excess baggage.

That said, my reason for replying is to note that in the US midwest, why as an interjection often is pronounced differently than why as an adverb; the h in the interjection being heard less than in the adverb. At forvo.com, which shows /waɪ/ IPA, I think the h is heard for about five of the pronouncers (eg 'TopQuark'), and not heard for the other four (eg 'migueld'). ('googler' might do both.)