What's the meaning of the phrase 'Bada-bing'?
'Bada-bing' is an exclamation suggesting that something is going to happen, easily and predictably - something like 'Hey Presto!'.
For emphasis, it is sometimes extended to 'bada-bing-bada-boom'.
What's the origin of the phrase 'Bada-bing'?
I don't think that I, as a native of England, had ever come across the phrase 'bada-bing' other than as the name of the strip club in The Sopranos. They used the spelling Bada Bing!. That name was chosen as a nod towards James Caan's character in The Godfather, who used the phrase repeatedly.
Not many in the UK would know it, but the phrase does have a meaning. It follows the common pattern of ablaut reduplicated phrases. That is, one word followed by a second word with the vowel changed. These expressions are often indicative of a twoing and froing action - like chit-chat or zig-zag. This suits the 'it was like this, then suddenly it was like that' meaning of the phrase.
The expression 'bada-bing' is Italian, as you might expect from the above references. It clearly pre-dates the 1972 Godfather movie and featured, as 'bada-bing-a-bada-bang-a-bada-bing' in the Italian-American Pat Cooper's My Hero comedy routine in 1965.