The ability to pass right-to-work laws are delegated by an amendment to the Taft-Hartley Act, passed by the US Congress over then-President Harry S. Truman's veto in 1947.
In a right-to-work state, the government makes it illegal for a business and a union to agree to a contract where one of the stipulations is that the employer will only hire union labor. This law only applies to labor unions, and not contracts with other corporations to provide labor. It is also illegal for a union to go on strike to prevent non-union workers from being hired. Thus, a union's right to freely negotiate a contract with an employer is denied by the government.
Right-to-work laws also make it much harder to organize a union. The ability of non-union employees to benefit from collective bargaining without paying dues creates a free rider problem, encouraging employees to leave (or not join) a union, making union activities less sustainable. Levels of unionization are typically much lower in right-to-work states, as are average wages. It is not clear whether low average wages are a consequence of lack of unionization, or whether both follow from a poor overall bargaining position for workers in these states.