McKinsey & Company is a global management consulting firm founded in 1926 by University of Chicago professor James O. McKinsey, that offers professional services to corporations, governments, and other organizations. McKinsey is the oldest and largest of the "Big Three" management consultancies (MBB), the world's three largest strategy consulting firms by revenue. The firm mainly focuses on the finances and operations of their clients.

Under the leadership of Marvin Bower, McKinsey expanded into Europe during the 1940s and 1950s. In the 1960s, McKinsey's Fred Gluck—along with Boston Consulting Group's Bruce Henderson, Bill Bain at Bain & Company, and Harvard Business School's Michael Porter—transformed corporate culture. A 1975 publication by McKinsey's John L. Neuman introduced the business practice of "overhead value analysis" that contributed to a downsizing trend that eliminated many jobs in middle management.
What is MBB and what does it stand for?
MBB is in reference to the "Big Three" management consultancies. From the passage, it isn't clear exactly what it stands for, but McKinsey is included in MBB as it is the oldest and largest of the group.