So, you have to pay dues, but you don't have to "join"? Ooookay. Why would anyone do that?
To keep their job. There's only one way a union can ever get you fired (which if you think about it, is most odd since the U represents the workers, not management). That reason is for not tendering dues.
My employment is covered by a union. I have no right to bargain over wages personally (it technically would be an unfair labor practice for me to ask my boss for a raise!). I don't represent myself, the union does. That said, I'm not a member of the union.
To keep my job, I must pay dues. I don't get to vote on the contract, or be an officer, but I will not be fined by the union if I ever speak out against it, or try to decertify it down the road. Unions have been pretty good lately at suing members who don't pay fines, and securing judgments against them.
Another potential benefit to not joining-- though I don't do it-- is to request to pay only "fair share fees". This would be some amount less than regular union dues (as unions can only charge non-members dues related to $ the union spends on negotiating and organizing, but not political activities).
This arrangement also lets people who are philosophically opposed to unions opt out. The right to refrain is as important as the right to form unions. But, that creates a free-rider problem-- a few non dues paying members would get all the benefits of collective bargaining, paid for by other employee's dues. To avoid the problem, SCOTUS has ruled that in a union shop, if you want to keep your job, you don't have to join the union, but you do have to pay at least fair-share dues.
The Taft-Hartley, though, gives any state the right to be more strict here, and so about half the states have passed right to work laws which remove union security. In a RTW state, the union can't force you to pay dues as a condition of employment. Unions aren't very strong in right to work states...
I think you might be slightly confusing the terms: In a closed shop, you can't even apply for a job unless you are already a member of the union. This is illegal (except in construction).
In a Union/Agency shop, anyone can apply for the job. Once you get hired, you have 30 days to join the union (or at least pay dues) or the union can get you fired. This applies only in non-right to work states though.