Besides the legal protections, the primary leverage a union has is the fact that in manufacturing, agriculture, etc., companies start hemorrhaging money if work stops. So a relatively short strike (and thus more easily weathered by the members) can have a massive impact. This worked even before there were legal protections for unions, because the union can strike faster than the company can hire scabs. Of course, prior to the protections, companies could decide that taking the hit was acceptable, and then just hire a bunch of scabs to replace the workers. Or they could threaten the union leaders with violence. Which of course lead to the unions using threats of violence against management and the scabs.
After a couple decades of increasing hostilities, accelerated by the re-introduction of a bunch of combat trained WWI vets back into the workforce, the US established a robust set of worker protections to eliminate the necessity of violence.