The NYT wrongly told readers that a bill approved by the New Hampshire legislature would, "disallow collective bargaining agreements that require employees to join a labor union." It is already the case that collective bargaining agreements cannot require employees to join a labor union.
Under current New Hampshire law, collective bargaining agreements can require workers to pay representation fees to a union. National labor law requires that a union represent all workers who are in a bargaining unit regardless of whether or not they opt to join the union.
This means that non-members not only get the same wages and benefits as union members, but the union is also required to represent non-members in any conflict with the employer covered by the contract. For example, if a non-member is faced with an improper dismissal the union is obligated to provide them with the same representation as a union member.
The new bill passed by the New Hampshire legislature effectively guarantees non-union members the right to get union representation without paying for it (representation without taxation). It denies workers the freedom of contract that they currently enjoy, which would allow them to require that everyone who benefits from union representation has to share in the cost of union representation.