"Why, thank you" is used when surprised or complimented. Because it is considered an old fashioned phrase it also adds a touch of humour.

Saying "why, thank you" presents a certain level of surprise and politeness. Because it isn't said very often anymore, you do need to be careful of the tone you take while saying it. Phrases such as this can come off as seemingly sarcastic if the politeness or surprise does not fit the situation. For example: Saying "why thank you" with a lot of emphasis on the "why" and a longer pause before the "thank you" after somebody does something they are expected to do. Hope this helps :)

"Why" is just a word used as a kind of "filler". It is actually an exclamation as far as I can tell when used that way. It equates to saying; "Oh." I would advise trying to employ it with any intention. It does not show one to be modest. It is just a sound. A simple "Thank you." is sufficient. It serves no useful purpose to adopt an affectation. It looks like a bit of an act, and people quickly tire of acts in public life. . There are affectations which, when adopted, make one the easy target of imitation and ridicule by others. Similar affectations such as Batting---the---Eyelashes, or moving the hand about in the air with some sort of flair, or turning the head away to show indifference are not tactful gestures in the modern world.