What does "why yes" mean?

Solution 1:

Why here is an interjection, placed at the beginning of a sentence to express surprise:

Is it nine o'clock already? Why, I must have fallen asleep!

or opposition:

Are you suggesting he stole the money? Why, I think that's impossible.

or in you example, emphasis:

B. Awesome, thanks!
A. Why [would you think it would be any less than awesome?], yes of course.

or perhaps

A. Why [would I even need to be thanked for something I'm happy to do], yes, of course.

Don't take the bracketed words as a literal ellipsis. The why is there to express a general emphatic tone.

The OED finds the interjectory use of why going back five hundred years.

Solution 2:

I speak Persian. And in Persian, we use the word why by itself as a synonym for yes in some situations, usually for emphasis. As the answer above suggested, it probably is the shortened form of "why not?" in both languages.