To get your carpets nice and clean you need 4 things . . .

1. a good wet/dry vac
2. a good carpet cleaner like Folex (available at Lowe's, Home Depot, and many other local stores)
3. some water
4. an upholstery brush

Basically, I vacuum my matts first to get out all the loose dirt. Then, I wet them down with a hose and spray them generously with Folex. I scrub the bejesus out of them, rinse them with the hose, and then extract the remaining moisture with my wet/dry vac. Leave in the sun to complete the drying process.

Voila! Clean carpets.

Don't put the matts back in your car until they are 100% dry. Some hack put matts back into my mother's old Audi when they were still wet and they got moldy. That's worst case, best case is they get mildew and smell bad.

 

^+1 great methods.

I forgot to mention the vacuum which should be a given. It's also good to first brush the carpet sweeping loose dirt into one corner and then vacuum. Much more thorough and faster. Try vacuuming the carpet. Then go back with the brush and you'll see all the stuff left behind. The dirtier the car the more it works.

I use the "dry clean" method I mentioned above and prefer a plain scrub brush without the traditional upholstery handle. That allows easy access around foot pedals etc.