You aren't being told the whole truth.
Sometimes web designers use templates they've bought in from someone else, or they use code which others have written. On such occasions there may be a condition attached to the use, which stipulates that an attribution link which appears in the code should not be removed. It's part of the copyright licence.
This may what has happened here, in which case you should have been told so when you asked. Either way, there's no such thing as links which can't be removed. I get intensely irritated when I hear of web designers trying to blind clients with science in this way.
It's not uncommon for designers to place links to their own sites in a discreet position somewhere on sites they've designed - it's a useful selling aid, and most clients are happy to agree to the practice. You don't say if the link in question is in this category, but if it is you might want to reconsider your request that it be removed.