Trying to understand reduced mass

I understand that the reduced mass is used to simplify a two-body problem into a one body problem, so that instead of calculating two masses rotating around a common center as in the earth and moon, you can treat the earth as stationary, and then used a "reduced" mass for the moon in the calculation.

My question is how is m1m2/m1+m2 calculated to derive the reduced mass. I'm not able to follow the arguments in my readings. Can someone give me a simple, straightforward derivation along with a conceptual "intuition" as to why this works?