Holacracy doesn't eliminate the functions of "managers".
Putting all the functions of a manager in one person is what is minimized.
The power is shifted to the group itself collectively. The group then delegates and distributes this authority to the roles in that group. Once a role is given the authority to decide something, that role then decides things autocratically, i.e. by him/herself.
So what you have is structures and policies decided collectively, and day to day operations decided autocratically, with as much speed as needed to get work done.
This can be changed at any time by the group, when any member feel that it is not working quite well or causing harm to the group itself, thus hopefully, gradually evolving towards better and better "management".
In one extreme, in theory, a group may choose to delegate all "managerial" functions back to a single role, which will make that role look just like normal "managers". However, when any one member feels like there is something that's not quite right with it, he/she can call a meeting of the group to change it, whereas normally, the role of a manager is given, and there is no proper and transparent way of changing it, especially by subordinates, even when there are better ways of "managing".
Holacracy is based on Sociocracy, which is a framework for decision making and organizational structure where no one can be ignored. It does not do away with hierarchy, but it does put a check on the power of bosses and owners. Real influence flows not just top-down but also bottom up. Therefore workers are more invested in the work and investors are more invested in the workers. Read We the People by John Buck and Sharon Villines. Check out websites sociocracy.info sociocracyconsulting.com or look it up in Wikipedia.