To what extent should the manager explain his decision to the staff? Should he always give the full background why he decided this? or just give a brief explanation to have his staff buy-in, or should he just inform the staff with his decision without explaining himself?

Manager should explain his decision to the extent ;  based on the management decision(s) or to the extent get the job done .  

Manager should develop a stronger relationship with your employees and worker feedback is critical for the productivity and get job done.

The manager can give a brief explanation to the staff, however, manager can solicit feedback and discuss current problems and possible solutions.