'Free City' with Babylon (B): How to play the special ability (2nd stage)?
The player controlling the Free City during the 6th turn plays the last remaining card in the Free City's draw pile.
The 7-wonders rules define each Wonder boards special abilities. (page 9)

The Hanging Gardens of Babylon the second stage gives the player the option of playing their seventh Age card instead of discarding it. This card can be played by paying its costs, discarded to earn 3 coins or used in the building of the third phase of the Wonder.
Clarification : during the sixth turn, the player can therefore play both cards they have in hand. If the second stage of the Wonder is not built, the Babylon player can then build it on their sixth turn and then play the seventh card instead of discarding it.

There isn't really a 7th turn in an Age, but it helps to think of it as an extra turn, since the resources used to build the 6th card as well as the 6th card itself are available to build the 7th and last card. This 7th card is normally the card that you would discard during the 6th turn of an Age, so that is the card that the player controlling the Free City can play to construct a structure, build their wonder, or if both if those are impossible to discard for 3 coins. (Page 7)

Clarification : The Free City cannot discard a card to get 3 coins unless it cannot play a card.
Note : during the sixth turn of the age, there is only one card remaining in the Free city’s draw pile. This last card is discarded at the same time as the seventh and last card of the players.