I came across this question today and the accepted answer starts with "Why yes, yes you can". Why didn't the person simply write 'yes, you can' instead of adding two extra words? Is this person trying to emphasize the obviousness of the usage of the word in the given question? What is the difference between them?
Why can serve as an interjection. From the Merriam-Webster Dictionary:
Why -- used at the beginning of a statement, especially to express surprise
Why, here's what I was looking for!
So you guessed right:
Is this person trying to emphasize the obviousness of the usage of the word in the given question?
Why, yes, of course!
As Merriam-Webster notes, this usage of why is "somewhat old-fashioned".
Why how now, Madam Flirt?
If you thus must chatter;
And are for flinging Dirt,
Let’s try who best can spatter;
Why how now, saucy Jade;
Sure the Wench is tipsy!
How can you see me made
The Scoff of such a Gipsy?
Another subtle difference in meaning between "Yes you can" and "Why yes, yes you can" is that "yes you can" is a straightforward statement of giving permission or judgment of ability of the listener to perform some action, while "Why yes, yes you can" is often used by a speaker who is trying to be humorous about a situation where it should be obvious to the listener that they shouldn't have needed to ask permission in the first place.
Q: Can I eat that food? A: Yes you can. Q: Can I fall down? A: Why yes, yes you can.