It literally means that they're available and willing to help you anytime, but it is usually used just as an alternative to 'you're welcome', just like 'no problem'.

It simply means 'at any time' the person is ready to do the same favor the other is thanking him for. It has become an automatic answer to 'Thank you' in many cases as Maddie mentioned.

I have the same idea with Maddie... "thank you!" "If you need my help, just call on me anytime." by simply replying "anytime", would mean the above phrase. :)