The move that sticks out for black is Rxd5+ pulling the king deeper into enemy territory.
The forced sequence is:
..., Rxd5+
Kxd5, Rd8+
Kc6, Qd7+
Kb7, c6+
Kb6, Rb8+
White cannot defend against mate at this point, with mating threats like Qb7, Qa7 and Qd5 regardless of where the white king moves next.
White could try to delay it by moving Ka6 instead of Kb6 but the position is exactly the same after ..., Ra8+, Kb6, Rb8+.
Edit: An alternative line that's also interesting,
..., Rxd5+
Kxd5, c6+
With the idea of opening up lines for the queen and rook.
You are right that 4. ... c6+ wins, but because of 5. Ka6 it is not the fastest possible win. :)
Actually 4. ...c5+ wins faster because after 5. Ka6 Qc6+ is possible, and after 5. Kb6 follows 5. ... Rb8+ 6. Kxc5? Rb5#.
The question stays how/if to win after
?
Now white's king have to shift between d and c-file. Moving king onto b-file is a mistake because of Qb4. Goal of black now is to get the queen to d5 square through tempo checks. Note that king cannot escape. For example, say king gets to d2 somehow, black can always give check with queen on d-file forcing king back onto c-file.
Once black has Qd5 and Re6, that's just a classic mating net, Rc6+ and Qb5+ etc. since the queen and rook are no longer tripping over each other. The moment black forces the white king onto b-file mate is going to follow.
There's also mate with c6 pawn if white doesn't play accurately.
Correct until here. I can tell you there is a much faster win than 4. ... Re6+ 5. Kc4 ...