They're referring to the Revised Directive on Payment Services ("PSD2") passed by the European Union which imposes a requirement to use "Strong Customer Authentication" for certain financial transactions (online or contactless) to reduce fraud, among other things.

When you make a payment over a certain size, you're required to verify that it's you making the payment. It's 2FA for payments essentially, you enter the code sent by SMS or you tap the approval button in your banking app, or enter your PIN again for contactless transactions.

It seems a lot of payment institutions have allegedly implemented SMS verification for these transactions. I bank with Monzo (https://monzo.com) which offers an approval notification in their app.

Unless you switch to a "challenger" bank like Monzo, you're going to be getting SMS to verify transactions (otherwise the transaction won't go through) and while I consider it hyperbole to mark this as "practically illegal", it does make things rather difficult for those with no phone or SIM card.