Correct, but they don't need to prove that the statement was untrue.
Imagine that you are lying MP who takes funds from arms dealers who put you up in a hotel, say the Ritz in Paris. If I say that the arms dealers picked up your hotel bill you can sue me for libel.
Given the transaction is between you the arms dealer and the hotel. It is very hard for me to evidence that it was not you that paid it.
It would be very easy for you to produce a copy of the bill and show the money coming out of your account. If the burden of proof was on you, that is what you would have to do to convince the court.
If the burden was on me, you could simply lie, say you wife paid it and stand behind the shield of justice waving your sword of truth. Despite the fact that you have provided no evidence the court starts off on your side.
If something is printed about you that is false, you should have to show that it is false as well as printed and damaging.