I have encounter this phrase in a few occasions but I couldn't fully understand the meaning of it.
In order to try and make this question more complete I searched the web for examples of this phrase:
From: HuffPost Politics
...his rhetoric is scratching the proverbial itch of Americans...
From: Working at Booking.com
Some of our most useful hacks inside the company have come from someone scratching the proverbial itch.
From: Detailgal Blog
My husband quit a solid job in the name of scratching the proverbial itch that hasn't gone away in over 10 years.
From these examples I figured that it should mean something around the lines of "solving something that bothers you", but I would like to confirm whether this is true and to what extent.