We have the same package as yourself, with line rental paid in advance, we don't use 1571, and were equally puzzled until a BT email five days ago:
In January, BT Answer 1571 is changing.1 You'll be able to keep more messages for longer and check them remotely. From 4 January 2014 this service will be £1.75 a month.
If you have BT Privacy with Caller Display it will cost you £1.75 a month from 4 January 2014. But you can sign up and get BT Privacy with Caller Display free for a year (a new 12-month line rental contract applies)
The way BT has been hiking its prices we aren't prepared to accept yet another year's contract which presumably applies after our existing one ends in August next year. Like so many 'free' offers BT's caller display has strings attached.