How does an F1 driver become slower with age?

I'm going to address the general question of what causes drivers to get slower with age, and not the specific example you mention (that of Vettel).

As Greg noted in the comments, an F1 driver's reflexes will degrade with age, but so will their physical ability to actually drive the car. Formula One cars are incredibly physically demanding to drive, regularly pulling loads of about 5g under braking and during cornering. It's not just the G-forces they need to be able to withstand, either: it takes a force of 125kg (about 350lb) just to operate the brake pedal.

With that in mind, it's no surprise to me that most drivers (with a few outliers) retire from F1 between the ages of 35 and 40. That's in line with most other professional sports, like association football and basketball. Even if a driver's reflexes remain as sharp as ever, there will eventually come a point where their body simply can't take the strain anymore.

It's worth noting that in the early years of the sport, when the physical demands were much different, experience was more important than fitness, and so it was common for drivers to remain competitive at much older ages. Juan Manual Fangio won his final World Championship at the age of 46 - in the process delivering what is widely considered the greatest F1 drive of all time - and Louis Chiron came 6th at the 1955 Monaco Grand Prix at the age of 55.