Although he is reported to have spoken his mind freely while being a

while is used in two senses
1) as a time frame
for example: I was watching TV while my girlfriend was making a sandwich. (the chauvinism is just for fun and laugh. dont frown)
2) as a contrast
for example: My girlfriend bought a louis vuitton bag while I am out of work.

This sentence, as answer choice B puts it, becomes something like:
I played fooltball while in childhood.
This does not make sense: it tries to be a concise version of: I played football while I was in childhood. Even the longer version seems awful.

Just to clarify:
There are tons of things the GMAT likes and doesnt like. The best way is to develop a kinship with what the GMAT prefers.
For example: webster defines the word "like" as : 1) resemblance 2)such as.
But you will definitely get it wrong if you use like as a substitute for such as in a gmat sentence.
We all know this.
How?
Because every book has written it.
However, a lot of books do not have stuffs like this which are less prevalent on the GMAT preference list. So its better to reply on practicing tons of official problems and develop the feeling yourself.
Hope this helps.
Souvik