Examples

White's back-rank is weak, because it is only protected by the rook on c1, and White's king has no escape squares.
Black cannot force checkmate with 1...Qd1+? 2.Rxd1? Rxd1#, because White would be able to defend the back-rank with 2.Qf1!, and the queen on f1 is x-ray defended by the rook on c1.
However, Black can exploit the weak back-rank with 1...Qxc4!, winning the queen.
If White recaptures the queen with 2.Rxc4, the rook is deflected from the back-rank, and Black can deliver back-rank mate with 2...Rd1#.
(Note that if it would be White's turn in the initial position, White could win with the same trick.)

Black's back-rank is unprotected.
With the last move ...Rd5 Black protected against back-rank mate, as against 1.Qb8+? (or Qc8+), there is now the defence 1...Rd8 (or Qd8).
Nevertheless, black's last move was a decisive mistake.
White can now win with 1.Qxd5!, exploiting the weak back-rank, and winning the rook.
If Black accepts the queen sacrifice with 1...exd5, the e-file is opened, and White can deliver back-rank mate with 2.Re8#.
On the last move, Black should have made an escape square for the king, for example with ...h6.

Sometimes the weak back-rank can be used to defend against your opponent's threats, as in the following example:

White already has a decisive material advantage, but with the last move ...Rc1 Black has defended against back-rank mate, is attacking the rook on c7 and threatens to promote the pawn on d2.
White can exploit the weak back-rank with the clever interference 1.Nc4!!, renewing the threat of Rc8#.
Now White wins in all variations:

• Promoting the pawn with 1...d1=Q does not prevent 2.Rc8#.
• 1...Rxc4 2.Rxc4 doesn't help because 2...d1=Q still allows 3.Rc8#.
• 1...Kb8 prevents checkmate, but White can win with the intermediate check 2.Rb7+ Kb8 3.Nxd2.
• Finally, the clever 1...Rg1+, hoping to decoy the king and promote with check after 2.Kxg1? d1=Q+ fails to 2.Kh3!, for example 2...Kb8 3.Rb7+! Ka8 4.Nxd2.