Please explain this reduced mass equation of electron? How electron looses its mass because of the oscillation of nucleus about its centre of gravity?

Reduced mass is the concept of introduction of pseudo force in the system. 

This concept is so broad that you can apply it to any situation where classical mechanics govern the system and there is 1 accelerated frame and motion of other is observed.
Example 1:
Let's say only Sun and Earth exist in solar system. Obviously, we will observe the motion from earth, which is our accelerated frame of reference. We know that sun will be revolving around the center if mass of Earth Sun system. What if you want to analyses the motion of sun without worrying about your own acceleration? Just write the answer taking yourself at rest and mass of sun as reduced mass of the system.
Example 2:
Reduced mass also works at subatomic particles. Iff velocity concerned is very low than that if light (5*10^5).
You can set electron as your reference frame and write equations of motion for nucleus using reduced mass.