"...they are both trying to turn inwards all the time..."

Not at all. Remember that they're both rigidly afixed to a rigid, solid axle.

But in your mind's eye, imagine what happens when the railroad car drifts sideways slightly. Now one wheel is riding on a slightly smaller diameter, and thus trying to go faster. The other wheel is now on a slightly larger diameter, and trying to go slower.

But both wheels are forced to rotate at the same speed because they're rigidly tied to a solid axle.

So the slower wheel is driven faster -- this has the effect of "steering" toward the center. Thus, whichever side drifts toward the outside, it becomes the steering side and the railroad car is nudged back toward the center.

You can also see that the inside wheel, which is on the smaller diameter and going faster, feels some drag from the slower wheel. Thus, it tends to steer toward itself -- which also brings the car back toward center.

This only works while in motion.

The flanges keep the car on the rails while at rest (while loading a boxcar with a forklift truck, for example). They probably also play a role going around tight curves.