How do you checkmate with a rook? How do you checkmate with a Rook and King versus King? Is there a good and fast method? Can you explain it in a good way? Thanks!
I know you're a FIDE Master :), so I suppose you're more interested in this question from a teaching perspective. 

The simplest way to understand a checkmate with King and Rook vs King is the idea of the rectangle of the opposing king. Consider this position- 



Here, the Black King is restricted by the White Rook in this giant rectangular area of the chessboard. For simplicity, I am not considering the area restricted by the White King, which would be like this - 

 

A simple way (I'll deal with optimizations later) to checkmate is to ensure that this rectangle gets smaller and smaller, so that the Black King is pushed to a corner of the board (more visual representation below with diagrams). 

    [FEN "8/8/8/8/4k3/8/8/K6R w - - 0 1"]

     1. Re1+ Kd4 2. Kb1 Kd3 3. Kc1 Kd4 4. Re2 Kd3 5. Kd1 Kd4 6. Kd2 Kc4 7. Re3 Kd4
     8. Ke2 Kc4 9. Rd3 Kc5 10. Ke3 Kc4 11. Ke4 Kc5 12. Rd4 Kc6 13. Rd5 Kb6 14. Kd4
     Kc6 15. Kc4 Kb6 16. Rc5 Kb7 17. Kb5 Ka7 18. Rc6 Kb7 19. Kc5 Ka7 20. Rb6 Ka8 21.
     Kc6 Ka7 22. Kc7 Ka8 23. Ra6# 


1. Re1+ Kd4 




2. Kb1 Kd3 3. Kc1 Kd4 4. Re2 Kd3 5. Kd1




5...Kd4 6. Kd2 Kc4 7. Re3 Kd4




8. Ke2 Kc4 9. Rd3




9...Kc5 10. Ke3 Kc4 11. Ke4 Kc5 12. Rd4




12...Kc6 13. Rd5




13...Kb6 14. Kd4 Kc6 15. Kc4 Kb6 16. Rc5




16...Kb7 17. Kb5 Ka7 18. Rc6 




18...Kb7 19. Kc5 Ka7 20. Rb6 




Ka8 21.Kc6 Ka7 22. Kc7 Ka8 23. Ra6# 1-0




Of course, with this method, the one mistake that White must avoid is stalemate - 


Rb7?? stalemate! 




Optimizations and Tricks. 

More experienced players can get to smaller rectangles quicker by looking further ahead. 

One trick is to trap the king in a smaller rectangle if it comes too close to the rook. For this, White also has to use the king more actively.  

   [FEN "8/8/8/8/4k3/8/8/K6R w - - 0 1"]

   1. Re1+ Kd4 2. Kb2 Kd3 3. Kb3! Kd2 4. Re4! Kd3 5. Rc4




Another trick is to give a rook check when the kings are in opposition. 

   [FEN "8/8/8/2k5/8/2K5/8/4R3 w - - 0 6"]

   1. Re5+!? (1. Rd1!) Kd6 2.Kd4




This trick is not necessarily optimal. In the above example, White can actually mate faster with the move 1. Rd1! 

   [FEN "8/8/8/2k5/8/2K5/8/4R3 w - - 0 6"]

   1. Rd1! Kb5 2. Rd5+! Kc6 3. Kc4




The above optimization is also made possible by the rook-waiting-move trick. If, in the above position, instead of going to c6, the Black king tries to escape via a4, then the following checkmate is possible. 

 [FEN "8/8/8/1k6/8/2K5/8/3R4 w - - 0 7"]

 1. Rd5+ Ka4 2. Rc5! Ka3 3. Ra5#


One more idea is to use king-opposition (in combination with the rook) to push the enemy king behind. 

 [FEN "8/8/8/2k5/8/8/2K5/4R3 w - - 0 4"]

 1. Kc3! (1...Kd5 2. Re3!) Kc6 2. Kc4! Kd6 3. Re4


Combining these different ideas, here are two of the most optimal mates possible from the starting position - 

   [FEN "8/8/8/8/4k3/8/8/K6R w - - 0 1"]

   1. Re1+ (1. Kb2 Kf4 2. Kc3 Ke5 3. Re1+ Kf6 4. Kd4 Kf5 5. Re4 Kf6 6. Re5 Kg6 7.
   Ke4 Kf6 8. Kf4 Kg6 9. Re6+ Kg7 10. Kg5 Kf7 11. Re5 Kg7 12. Rf5 Kh7 13. Kf6! Kg8
   14. Rh5 Kf8 15. Rh8#) 1... Kd4 2. Kb2 Kd3 3. Kb3! Kd4 4. Kb4 $3 Kd3 5. Kc5! Kd2
   6. Re4 Kd3 7. Re5 Kc3 8. Rd5 Kb3 9. Rd3+ Kc2 10. Kc4 Kb2 11. Rd2+ Kc1 12. Kc3
   Kb1 13. Kb3 Kc1 14. Rd3 Kb1 15. Rd1# *