Anyone know why all the knobs, light switch, radio knobs, turn signal, etc. become sticky?

Jelly fingered kids?😕

On one of the TV car restoration shows last year, someone had an expensive sports car with the same issue.  I believe they cleaned the knobs and repainted them.

Age, sunlight, heat, skin oils, etc. They all contribute to the deterioration of plastics. Sometimes you can clean the stickiness off but if it's been there for some time, it's probably penetrated below the surface layer.

👆 What he said. The steering wheel on my '74 got to be unbearably sticky. I used 3M Adhesive remover to wipe it clean, and it worked great, with no damage to the wheel at all. Any "soft" plastics like this will eventually get sticky, as the chemicals used to form them ooze out over time. I have a Corvette refrigerator magnet on my toolbox that is sticky like that, Has been for years.

I had the same problem on the entire inside of my car.  In my case it was from 50 years of using ArmorAll to brighten everything.  I tried everything to remove it with no success.  At the Good Guys Summit Racing Nationals car show in Columbus this year I spoke to the Jax Wax people and found out that ArmorAll has silicone in it and removing requires something to break down the silicone. I've got about half of the car done by using mineral spirits, followed by Totally Amazing, following with a citrus cleaner. I know this sounds harsh, but it works and there is no damage to any of the surfaces I have used this method on. 

I second Joel’s recommendation of 3M Adhesive Remover. When I bought my 74 16 years ago the steering wheel was very sticky. Cleaned with adhesive remover and hasn’t come back in 16 years.