Can someone explain this new Mass law?

So what is the "procedure" when an illegal invader checks both of those boxes, whether by mistake or intentional?

• It is all right there in the AP article you cited but (apparently, naturally) failed to read:
  

"Under the current process, the RMV provides the secretary of state’s office with all the relevant information for voter registration -- such as an applicant’s name, date of birth and address -- and can provide additional information to further verify voting eligibility,said Debra O’Malley, Galvin’s spokesperson.

“They send over a batch of registrations each night to our office, which are then distributed to the appropriate cities and towns for local election official review and certification,” she said by email. “The RMV has a record of what evidence of lawful presence has been provided and removes from those batches anyone who hasn’t provided them with a U.S. birth certificate, U.S. passport, or U.S. naturalization papers.”

• Also worth noting this detail:
  

Additionally, 16 other states and the District of Columbia have enacted similar laws and have not reported non-citizens being illegally permitted to vote

• Several states have actually made clerical errors that registered non-citizens as voters or encouraged non-citizens to vote. 
  
• Some pretty good research of the 2008 and 2010 elections suggested that 1.5-3% of non-citizens self-reported voting in an election which certainly suggests that the number is larger than zero.  Still, actual reported instances of any non-citizen vote in 2020 is zero, as far as I can tell.
  
• Federal law started prohibiting non-citizen voting in 1996 but exempted non-citizens who had good reason to believe they were eligible.
  
• Every State prohibited non-citizen voting by 1926 but many local jurisdiction retain the authority to decide who votes in local elections and some have recently permitted local voting by non-citizens.

and can provide additional information to further verify voting eligibility,said Debra O’Malley, Galvin’s spokesperson.

“They send over a batch of registrations each night to our office, which are then distributed to the appropriate cities and towns for local election official review and certification,” she said by email. “The RMV has a record of what evidence of lawful presence has been provided and removes from those batches anyone who hasn’t provided them with a U.S. birth certificate, U.S. passport, or U.S. naturalization papers.”

Still, actual reported instances of any non-citizen vote in 2020 is zero, as far as I can tell.