B

Step 1: Understanding Escape Velocity
The escape velocity from the surface of a celestial body can be calculated using the formula:
$$ v_e = \sqrt{\frac{2GM}{R}} $$
where:
- $v_e$ is the escape velocity
- $G$ is the universal gravitational constant
- $M$ is the mass of the celestial body
- $R$ is the radius of the celestial body.

Step 2: Comparing the Moon and Earth
1. The mass of the Earth ($M_{Earth}$) is significantly greater than the mass of the Moon ($M_{Moon}$).
2. The radius of the Earth ($R_{Earth}$) is also greater than the radius of the Moon ($R_{Moon}$).

However, the key factor influencing escape velocity is the ratio of mass to radius in the formula. While the Earth has a greater mass and radius, the radius of the moon being smaller leads to a lower escape velocity value when plugging into the formula.

Step 3: Analyzing the Options
Option A: Moon has no atmosphere while the earth has.
This statement is true, but it doesn't directly affect the escape velocity formula. It may affect the conditions of leaving the surface but not the mathematical calculation of escape velocity itself.

Option B: Radius of moon is less than that of the earth.
This is true and directly contributes to a lower escape velocity as shown in the equation above.

Option C: Moon is nearer to the sun.
This is incorrect as the distance from the Sun does not play a role in the escape velocity from the Moon or Earth.

Option D: Moon is attracted by other planets.
While this statement is somewhat true, it does not impact the escape velocity derived from the Moon's own mass and radius.

Conclusion
Thus, the correct answer is that the escape velocity from the Moon is less than that of Earth primarily because of the smaller radius of the Moon.
Therefore, the correct option is B.