How do subscription services get my updated payment info without me entering it in?

Hello All,

I canceled my debit card through wells fargo in an attempt to stop all of the subscription services that I had signed up for. I got a new debit card number but I checked my statement today and it shows that I still got charged for Netflix, GoDaddy, etc. I haven't even bought anything with my new card yet!! On the statements it shows a charge at the card *---* where as all of my other charges show *--***-Last Four Digits.

How do these companies get my updated payment info without me entering it in and how can I stop this from happening?