The black king could have headed towards the square e8: 2. ... Ke7 3. Rh7+ Rf7 (3. ... Ke8 now loses to 4. Re6 mate) 4. Rb7+ and Black will lose a rook.
Position after 2. ... Kc7
The only difference between this position and the start position is the rook has moved from b1 to b6 and it's still White's move. The Rook is on the sixth rank for two reasons: (1) to shield the white king from lateral checks by the black rooks and (2) to form a barrier that the black king can't cross after the other rook checks on the 7th rank.
3. Rh7+ Kc8
4. Rc6+ Kb8
5. Rb7+ Ka8
6. Rcb6
And Black can't avoid mate next move by 7. Ra7 without giving up a rook.
Final Position: Black cannot avoid mate.
The immediate threat is 7. Ra7 mate. However, even here White must be careful. The Black King has no moves and this always hints at stalemate tricks. Here all Black has to do is give away both rooks for a stalemate. White has to be wary enough to avoid it. Let's play on a bit from the "Final Position". 6. ... Rd7 7. Rxd7 Rf7
Position after 7. ... Rf7
Now taking the "free" rook results in stalemate. But
