What is the reasoning behind the Hill Sphere? According to Wikipedia, Hill Sphere is: the volume of space around an object where the gravity of that object dominates over the gravity of a more massive but distant object around which the first object orbits.
True as this may be, it just mathematically supports a phenomenon that has been observed but it does not give reason or logic as to why does this happen in the first place. I mean why should the gravity of a less massive object dominate the gravity of a more massive one?
I wasn't aware of the Hill Sphere until recently when I was trying to visualize the orbits of different celestial bodies. The Hill Sphere comes closest to explaining why the moon orbits the Earth, more than it orbits the Sun and why the Earth orbits the sun, more than it orbits the center of our galaxy. By this logic all celestial bodies within the Gravitational pull of the center of our galaxy should directly be orbiting the center.
My argument is that if the Hill sphere of the Sun is as large as the solar system itself, any object within this sphere should be orbiting the sun. Why was the moon caught into the earth's gravitational pull in the first place when it had a much stronger pull from the sun? 
The answer to this would also eventually clarify why the earth orbits around the sun and not the center of the milky way.
 A: The moon orbits around the sun, but so does the earth. They orbit together with the moon's orbit perturbed by the nearby earth. If fact, despite their different masses they experience the same acceleration, so it shouldn't be surprising that they are bound to the same orbit since they are bound to each other (i.e. at basically the same distance from the sun). 
The moon experiences motion relative to the earth and is bound to it by the earth's gravity, and once bound, unless the tidal forces due to the sun pull them apart, they will stay bound together - accelerating towards the sun at the same rate, as essentially one object. This is the key: outside of the Hill Sphere the difference in gravitational force (the so-called "tidal force") is great enough to break the gravitational binding.
You may be interested to know that the Earth-Moon Roche Limit vis-à-vis the sun is about 33.6 million kilometers and that the earth is roughly 150 million kilometers from the sun. So we are quite safe from the danger of having our moon stolen.
A: 
I mean why should the gravity of a less massive object dominate the gravity of a more massive one?

Within the Hill sphere of the Earth, objects can orbit the Earth, because in the non-rotating frame of reference centered in the Earth (moving with acceleration around the Sun, so the frame is non-inertial), the Sun's gravity force is for the most part cancelled by the inertial centrifugal force. The remaining weak force is called $tidal~force$; it is negligible near the Earth, but gets stronger farther from the Earth.
The radius of the Hill sphere $R_H$ can be roughly estimated from the condition that for object at this distance $R_H$, the tidal force due to the Sun (increasing with distance) is already strong enough to counteract the attractive gravity force due to the Earth (decreasing with distance).

Why was the moon caught into the earth's gravitational pull in the first place when it had a much stronger pull from the sun?

We do not know for sure how Moon came into being Earth's satellite. One theory says Moon is a former part of the Earth, ejected after impact of some foreign body on the Earth's surface. Or it could have come from the outer space and by chance came close enough to Earth with low enough velocity that it got captured (the gravity due to Earth does not need to be stronger than the gravity due to Sun for that, as explained above).
A: "I mean why should the gravity of a less massive object dominate the gravity of a more massive one?"
Maybe because it's closer?
"The strength of the gravitational force between two objects depends on two factors, mass and distance."
