Basically:

• Go to Free Play
  
• Select any options (I prefer Seasons - Track Day Hyper Cars for speed purposes, but any category will do)
  
• Make sure you have 1 Round in Race mode with a Grid Size of 1.
  
• Any track (I recommend Crescent Valley Oval for quickest getting of the miles, or Okutama Grand Circuit (A or B) as a secondary recommendation)
  
• 99 Laps!
  
• The rest of the options can be whatever you like.
  
• Enter the race, yada yada, 3, 2, 1, GO
  
• Head to the first checkpoint on the map (On the mini-map it will be a white line on the circuit that you will drive through, for Crescent Valley Oval it is after the first turn, and right before the second one)

(If you're having trouble seeing the white line on the mini-map, just wait until you see the purple split arrow and time show up for the first time.)

• Quit using the Pause Menu, using the Quit option (not retire!)
  
• Then once you load back into the menus, Alt+F4. (You can exit the game by going through the menus too, but this is very slow, so I recommend just Alt+F4'ing it or using Task Manager if Alt+F4 isn't working for you.)
  
• Boot the game back up again
  
• Go back to free play, back to your track, go right back into the race again with no settings changed.
  
• Drive to the first checkpoint (white line) again.
  
• Bam, your achievement progress should pop up with more miles on the progress bar (around 300-600) (I've gotten almost 1000 sometimes)

You can continue to string these glitches by driving to the first checkpoint, quitting, exiting the game, entering back in, driving to the first checkpoint, repeat, and your miles should keep going up.

EDIT: It seems the specifics of methods and the amount of miles you get each time varies from person to person. Feel free to experiment like some of the commenters here have.

Troubleshooting:
If the glitch isn't working for you:
A. You ♥♥♥♥♥♥ up somewhere.
B. Codemasters patched the glitch.

Figure out which one is true for yourself.

With this glitch, it'll probably take you still a bit of exiting and entering the game to get to 40,000 miles or so, but it's still faster than doing 99 laps of Crescent Valley 99 times. What a stupid achievement. Have "fun".