Working for someone else is an agreement: I will provide a service for you in exchange for money. That agreement is ongoing. When one side or the other feels that the bargain is being violated in some way, fairness dictates the two should sit down, discuss, and reach a solution that is fair to both sides.
Your boss or supervisor doesn't come to the table as an individual like you, but with all the resources the company can muster: lawyers, researchers, consultants, etc. Most Americans believe in justice and fair play. They feel that a level playing field is a good thing. Businesses above the "small business" level have resources to enforce their position on a disagreement far beyond what an average employee can muster. The only way a fair resolution can be achieved is if you have access to equal backing. One guy against a corporation is a stacked deck, loaded dice, a blowout. When employees organize to form a unit, it goes a long way to making sure that both sides get heard.
I bet some of those who said that labor laws now make unions moot would also state in another forum that the less government regulation the better. Unions are not government. They are private, just like companies. Unions are a private sector solution.
Unions are no better or worse than corporations, their are good ones and bad ones, an lot somewhere between. Unfortunately, the ones you hear about are the usually the bad ones.
As far as the sentiment, "if you're not happy then leave" goes, imagine if that was extended to another bargain--that between a citizen and his government. "If you don't like a law, or a ticket, or a policy, then go to another country." That wouldn't be fair. We have mechanisms in our system to address grievances: elected representatives, the courts, etc. We don't say if you don't like it, lump it. Why should our working lives not play by the same rules? If someone paid a contractor to build you a house addition and he didn't finish it, or used cheap substitute materials, should the contractor be permitted to say: "If you don't like my work, go somewhere else? If you say, "no, you can go to court," then I say yes, unions make it possible for workers with few resources to go to court.
We are a government of checks and balances. Unions and corporations check and balance each other. The first amendment to the Constitution give you the right to organize for redress of grievances. To deny citizens the right to organize themselves in the workplace is un-American in the most fundamental way.
Sorry, but in fairness I had to speak out. There should be some balance in this discussion.