Attraction between Planet and Meteoroid:


Will it or won't it? 
Impression of a Meteoroid approaching Earth
Photo from TIME Magazine (Denis Scott / Corbis)

So now we know where the asteroids come from, but how do they move towards a planet as they do?
 
Mostly, it's Newton's Law of Universal Gravitation at work.
Basically, it says that every thing has an attraction to another object because of gravitational force.
This gravitational force is defined by: F=GMm/r2 .
The M and m are the two different masses in question while r is the radius (distance) between them.
G is the gravitational constant, which is: G=6.667X10-11 Nm2/kg2.
This means that the attraction between any two objects greatly increases when the masses are large and when the distance between them is small.
Note that the gravitational constant is small, and so it isn't seen as well on a smaller scale.
So as a meteor comes closer to a planet, the pull between them greatly increases.

Like planets, asteroids move in orbits that are similar to the orbits of planets around the sun.

roughly
Rough Diagram by Me.

They follow Kepler's Laws, which are:
                    1. All planets move in an elliptical orbit around the focus.
                            Note that the focus is not necessarily at the center.
                    2. A hypothetical line joining the satellite and the focus sweeps out equal areas in equal time intervals.
                            In other words, the closer to the focus the object is, the faster it travels.
                    3. The square of the planet's period is directly proportional to the cubed semimajor axes.
                           The period (T) is the time the satellite takes to complete the ellipse.
                            Kepler's 3rd Law comes out as: T2=(4pi2/GM)r3. M is the mass of the focus object.
There are things that can knock asteroids out of their orbit, however,
such as an impact with another asteroid, going near a magnetic field, or gravitational force.
These are the sorts of things that send asteroids towards another space object, such as Earth.

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