why are varmint barrels so much thicker then other rifle barrels, even though it shoots a smaller bullet?

A thicker barrel well flex less which in turn produces greater accuracy. That's the general idea but I'm sure someone will say their pencil thin barrel outshoots every bull barrel rifle they've seen but .....

3 reasons that I can think of
1 Yes less barrel flex makes for a more accurate rifle.
2 The barrel takes more shots to heat up and afect accurey
3 A heavy gun is easyer to shoot accuritly ,espeacly off sand bags

Here is some good reading.

http://www.chuckhawks.com/rifle_barrel.htm

As others have mentioned, thicker "heavy barrels" have less flex, so they don't whip as much with the bullet in the bore. The extra weight of the rifle also increases shootability and shifts the balance point of the rifle towards the muzzle (or actually towards the middle of the bbl, but it moves it towards the muzzle from previous position), so the muzzle doesn't hop as much on a shot, and is easier to stabilize while aiming.