First off, a pH of below 7 is actually preferred by a good 70% of commonly kept tropical fish. So I wouldn't worry much about it unless you have the sort of fish who prefer the more alkaline water. Who is living in the tank right now?
Now as far as the tap water, there is often a good amount of dissolved gas when it first comes from the tap. As this dissipates the pH can be affected, either going up or down depending on the gases that are released. (This is known as "out-gassing," and can be observed as lots of tiny bubbles in your buckets or bottles after the tap water has been sitting in them undisturbed for a few days.
The next thing is that biological action, more specifically the nitrifying bacteria, will naturally produce acids as by-products. That is why established tanks tend to go acidic over time.
You could also have other factors such as ornamental driftwood which would leach tannins into the water and cause the pH to drop.
Bottom line is, as long as the pH in your aquarium remains steady and above 6, you have no need to worry about it. In fact, as I said before, if you are keeping fish such as tetras, angels, rams, gouramis, bettas, rasboras, danios, barbs, and many others, this pH of between 6 and 7 is actually a great benefit to their overall health and happiness.
EDIT - I neglected to read that you are keeping mainly livebearers. (I thought it was your signature) So hmm...yeah you'll want to up that pH a bit. It has everything to do with buffering.
