What does the "starter" do in bread recipes such as ciabatta?

The starter gives the yeast time to activate and grow as yeast is the primary rising agent. The longer the yeast activates for the more flavourful your bread will be. I like to have my bread rise slowly, in the fridge for 12 to 14 hours and then again, on the counter, for another 5 or 6 hours. Makes for very tasty bread.

@tranquilsea I understand that, but I was wondering why does this recipe combine two doughs – one that has been left for one day and another that has only had 4 hours to rise?

As @tranquilsea says, the starter is for the yeast to rise. The need for starter for leavening is the reason the biblical Hebrews baked the unleavened bread before the first Passover.

I would say that the starter yeast is the primary flavour source, although it will provide leavening along with the second batch of yeast.

If you truly want to understand the difference…prepare the recipe both ways. Then, there will be no doubt in your mind. Let us know.