Imagine a seesaw with two children , one on each end with the same weight. The seesaw is balanced. Now imagine you swop one of the children with a heavier child. The balance isn't equal and the seesaw tips or swings towards the heavier child and that child has the advantage.
So if you replace the children with publicity on one side and none on the other, the swing goes to publicity.
So in your phrase, if you have publicity, you will gain an advantage or it will swing in your favour.