Why does it continually mention 'public sector' unions, specifically? Are unions lime the UAW still allowed to continue the practice?
Yes. The case does not affect private sector unions because there is no "state action" that implicates the Constitution when you're talking about private sector labor agreements. And trust me on this, I'm a labor lawyer.
Some states have "Right to Work" laws that permit employees to opt out of joining unions and not pay anything. Such laws usually (but not always) apply to both public and private sector unions. Without a Right to Work law, private sector employees can be required to join the union.
Public sector law is a bit different. Employees can never be required to
join the union, even if there is no "Right to Work" law. However, employees who chose not to join the union still could be required to pay a "fair share" fee (usually around 90%) , which is supposed to be that percentage of union dues that goes only to representational activities, such as negotiating contracts. The purpose of that was to prevent the employees from being compelled to contribute to a public sector union's political activities. Truth is, there was always a shitload of overlap. Anyway, because public sector employees could be required to pay that fair share fee, most just joined the union anyway. And that is what has changed.
What this decision said is that
public sector employees cannot be required to pay even this fair share fee. They can opt-out completely from paying anything at all to the union. This is really huge politically, because public sector unions are the most powerful of all Democrat constituencies. The result of this is that those unions will lose an awful lot of membership over time.