Kepler published his laws of planetary motion in the 1610's, establishing a heliocentric world model that was more consistent with observations than any previous one. Newton published Principia - which explained gravity - in 1687.
In between we have more than half a century where it's known that the Earth orbits the sun, but we have absolutely no way of explaining HOW HUMANS AREN'T THROWN OFF EARTH'S SURFACE from the motion. Was this question ever even asked at the time?
Let me start by saying that Newton didn't explain gravity. Indeed, he explicitly refused to explain it: "hypotheses non fingo". In translation,
I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.
Newton presented a mathematical model of gravitation. This was not an explanation, but something potentially more useful: a quantitative way to predict phenomena, even in the absence of understanding why. This is still largely the case - today we have two mutually-incompatible theories of gravity (general relativity and gravitons), and even if we consider them "explanations", we don't know if either is correct. In practice, this doesn't matter, since gravity works even if we don't understand why, and we can quantitatively model the effects of gravity (better than in Newton's day, since general relativity is a more accurate mathematical model of gravity).
Beyond this, it simply isn't necessary to understand the "why" of gravity to understand the effect of gravity that is relevant to your question: things experience a downward force. (This is not the only effect. First, the earth experiences an upward force, equal and opposite to the downward force acting on the object in question. Second, there are tidal effects. These don't matter as far as the question of why the motion of the earth doesn't fling things off.)
In Galileo's Dialogue on Two World Systems, the effect of the daily rotation of the earth is considered. Galileo's answer is simply the relativity of motion, and he gives the example of a moving ship. On a moving ship, a drop object falls straight down, relative to the ship, because it shares the forward motion of the ship. Whether the earth is rotating on its axis, or orbiting the sun, gravity will continue to act downwards (locally, relative to the surface of the earth). The basic observation of the effects of gravity - that things are attracted downward - is enough to explain why the rotational or orbital motion doesn't fling things off. Simply, gravity is sufficient to overcome the effect.