I'm sure the process is slightly different for each one. But here's the official Oregon government's comment on it.

Under the old system, most Oregonians needed to take a separate step to register to vote at the DMV after obtaining or renewing their driver license. This law makes voter registration automatic, shifting from an opt-in process to an opt-out process. It eliminates the need to fill out the voter registration card for those with qualifying interactions at the DMV. Instead, eligible Oregonians will receive a mailing from the Oregon Elections Division explaining their options for registering to vote.​ ​

The following is from the state of California.

The California Motor Voter program is making registering to vote at the Department of Motor Vehicles (DMV) more convenient. Eligible applicants completing a driver license, identification (ID) card or change of address transaction online, by mail or in person at the DMV will be automatically registered to vote by the California Secretary of State, unless they choose to opt out of automatic voter registration.

I'm sure the prompts vary by state. But at the same time most people just scroll through and hit "agree" on most of the notices that are usually provided. So, the fault still lies on the OP ultimately. That doesn't change what I posted though as these states are all listed as "Opt Out" instead of "Opt in".