They probably explained it mostly through Aristotelian Physics, which was only fully repudiated in the 17th century. Aristotle claimed that there can be no effect (e.g. motion) without a cause. He explained gravity by appealing to the nature of objects. "Earth" and "water" were heavy elements, so they naturally moved down towards the center of the Universe. "Fire" was a light element so it naturally moved up towards the heavens. There was no conception of forces or energy or anything that we might describe as modern mechanics. It even implies that gravity is dependent on an object's weight, which Galileo would prove to be false centuries later.

There seems to be some developments of this theory during Roman times. In De Architectura, Vitruvius counters Aristotle's claim that an object's weight determines the effect of gravity on it by claiming that mercury can hold up a very heavy stone, but let a light piece of gold sink through it (recognizing the existence of specific gravities). He again related this to the "nature" of the object rather than things like balancing forces, density, and pressure.

In all, Roman engineers, with some adjustments, probably explained gravity by using Aristotle's theories of natures, so water, being a heavy element, would flow down towards the center of the universe (towards Earth). They could utilize this fact and set up aqueducts to make water flow to where they wanted.

They didn't need to "explain" it — that's a very different perspective than a Roman engineer would have had.

Roman engineers were not scientists and did not need "science" to build aqueducts. They knew how to build them from rules of thumb developed by engineers over very long time periods. They were not searching for explanations about how the natural world "worked." Frankly they would not have gotten much use from anything a philosopher could have told them about this anyway; even through the 17th century and the "Scientific Revolution," the science of statics was still inferior to things figured out from practical engineering experience. In the 18th and esp. 19th century, this starts to change, but you don't need a concept of gravitational force to build an aqueduct. It was not until the 17th century that people really understood on a theoretical level why arches even worked.

If they did care about the "why" question, the reigning theory of the data was Aristotle's, and it basically said that all natural things have a natural motion, and the natural motion of water and earth is to move downward. There are no forces.

But again, this puts the cart before the horse. You don't need to know why water moves downhill to utilize that fact. It's pretty obvious that it does — if you asked a Roman engineer, "why does water move downhill?" he'd probably think you were a dolt. "It's just what water does, dummy. That's its nature. Every child knows that." The idea that one should quantify these kinds of "laws" and put them into an algebraic notation does not really show up until Descartes, and even then "why" questions are still pretty problematic (why does gravitational force obey an inverse square law? Even Newton had to say, I don't know — Hypotheses non fingo — I guess it's just its nature).