Q:

What does self organizing Scrum team mean?

What does the "self organizing" in "self organizing Scrum team" mean?

A:

See this article for a good explanation. This quote explains the heart of it:

Self-organization does not mean that workers instead of managers engineer an organization design. It does not mean letting people do whatever they want to do. It means that management commits to guiding the evolution of behaviors that emerge from the interaction of independent agents instead of specifying in advance what effective behavior is.

The team approaches a project, and, based on the project at hand, decides how best to allocate its resources to take advantage of each team member's various strengths.

A:

What does the "self organizing" in "self organizing Scrum team" mean?

It means this: No one – not even the ScrumMaster - tells the Team how to turn Product Backlog into increments of shippable functionality. The Team figures this out on its own. Each Team member applies his or her expertise to all of the problems. The synergy that results improves the entire Team’s overall efficiency and effectiveness.
Reference: Scrum Guide

A:

A self-organising team is one in which every member of the team is working towards some shared goal. Team members collaborate to reach the goal, valuing the team's output over their own individual productivity.
A self-organizing team tends to be:

trusting of each other's motives and motivation
helpful, respectful, communicative and transparent
collaborative with other teams
interested in delivery and feedback
goal-focused instead of productivity-focused.

The project manager's roles in a self-organising team tend to be:

making sure that the team has people with sufficient skills to succeed
helping them be aware of and understand information coming from elsewhere in the company
reporting the team's progress, successes, risks and obstacles to other audiences
getting stuff out of the team's way wherever possible (see Servant Leadership).

A self-organizing team is usually pretty good at getting things done, blasting through obstacles, communicating issues outside of their control and generally provides a great project experience for anyone using them. See also: Forming, Storming, Norming and Performing - you're looking for that last one.

