The main point of objection by the boycotters is that the referendum essentially does not accomplish anything. It is not legally binding and would not significantly advance the movement for statehood or independence; therefore, opponents feel that boycotting it emphasizes its ineffectiveness.
Part of the rationale for the opposition parties is that Congress is unlikely to take any action. For one thing, Congress ignored the previous referendum, where over 61% of Puerto Ricans who voted called for statehood. Many ballots were left blank, and government officials claimed that this meant that the results could not be trusted. Nevertheless, many Puerto Ricans do not believe Congress would do any better this time, in part because the referendum is non-binding.
The boycotters assume, largely, that Congress won't do anything. It would still seem that voting couldn't do any harm. However, many oppose the very idea of the referendum, either on principle or practicality:
- The referendum cost somewhere in the range of $8 to $11 million dollars - a symbolically sizable cost for an island mired deep in debt. Opponents argue that the money should have been used for something else.
- The fact that the referendum is not binding has led some party members to call it "inconsequential"; they feel that voting would give the referendum authority.
- Boycotts had been called for many weeks in advance. Originally, there were only two options on the ballot: independence or statehood. The Department of Justice ordered a third option that better suited many Puerto Ricans, namely, continuing their current state. The DOJ did not have time to fully review the final options.
- The ballot itself uses the word "decolonization" - another indication, some say, that the referendum is not consistent with the attitude of Puerto Ricans toward their current status.
The above meant that many Puerto Ricans saw the referendum as biased towards statehood, especially given that the governor, known as strongly pro-statehood, was the figure pushing it. The New York Times article you cited noted that referendums in the past have often been criticized as biased towards one side. Therefore, boycotting let the opposition parties challenge the process.
In a nutshell, boycotters see the referendum as an ineffective, financially-draining, and biased move on the part of those supporting statehood. By boycotting it and being able to claim that the result simply doesn't represent the opinions of all Puerto Ricans, they can criticize it and try to nullify the result. It's unclear if they would have had a chance to gain more than 50% of the vote between the other two options; previous polls showed that the option for statehood had only around %52 of the vote.