[IAmEveryone]

The article uses "beyond reasonable doubt" which is obviously a misapplication of the legal term of art. This being civil law, not criminal, the 50:50 proof, i. e. "preponderance of the evidence" should suffice.

In reality, I doubt there will be a practical difference to how it has been handled in the past here in Germany, where similar law has long been practice.

That means: your sign-up page needs a checkbox, that cannot be pre-checked, and that clearly states that it's an opt-in to receive these mails. This needs to be separate from any acceptance of ToS or anything else that is necessary for the transaction in question.

To verify the form submitter's identity, send a verification to their e-mail address (if you haven't already). Make sure the verification email does not already contain any advertisement itself.