why not escaped moon from earth to the sun?

For the folks not doing the calculations, the question is correct. The total force from the sun is greater than the earth (the sun is really big).

Sun-moon force ~ 4.4x10^20 N

Earth-moon force ~ 2.0x10^20 N

But that doesn't matter because the sun is pulling the earth at the same time.

Imagine that an astronaut in the space station had a 1kg object with a light string. He can twirl the object around himself by applying a force of 1 N to the string. But the earth is pulling on the object with a force of more than 8 N. How come the earth doesn't yank it from his hand?

Because everything is "falling" to the earth together. The object, the astronaut, and the station. So the absolute force of the earth is cancelled. The "orbit" can be maintained with no problem.

Likewise, although the sun's absolute force on the moon is greater, because it's pulling the earth at the same time, the moon's orbit around the earth is stable.

The Earth and moon could be thought as having a special case of a 1:1 orbital resonance, where the Earth and Moon alternate their distances to the sun (the Moon just moves farther than Earth). Picture two oval orbits, but in opposite directions from the sun. The planets in those orbits stay on the same side of the sun, but one would move closer while the other would move farther out, but would appear to orbit each other. These imaginary planets would orbit each other only once a year. Since the Earth and moon's orbits are more circular than the example planets, they are always close enough to tug one another to the point of the Moon appearing to orbit the Earth nearly 12 times a year.

The gravitational force on the moon from the Earth is stronger because the moon is *far* closer to the Earth than the Sun. The effect of gravity drops as you move further away from the source.

The sun is 400 times farther from us than the moon. Did you not know that?

I think that gravitational force falls off at the cube of the distance, which means that the sun's gravitational influence is reduced 64 million (400 cubed) times compared to earth, which more than takes care of the fact that the Sun is "only" 300,000 times the mass of the Earth.

For the same reason why the Earth doesn't drift closer to the sun. We're in an orbit, kind of like a dance that keeps us at a healthy distance.

I could try to answer this question but this medium is not appropriate to do so. Instead i'm going to link you to a picture on wikipedia that does well enough. Keep in mind, its a bit tricky to understand. And the moon's the easy one. Theres a lot more bizarre orbital paths out there, thats for sure.