Why does the force push it perpendicular to the direction of the velocity?
This is the most intuitive explanation I can think of:
When turning the wheels, the car still wants to continue straight ahead. Because of the inertia, the car is not just stopped but wants to go on forward (the effect is known as the centrifugal effect.)
When you turn the wheels, so they don't follow the motion perfectly anymore, they should slide over the asphalt as the car continues forward. You have then caused a velocity-component perpendicular to the wheels, in which direction they can't turn - they can only slide.
But the car doesn't start sliding and burning your Goodyear rubber tires (it doesn't continue forward without change). Static friction will prevent that. That static friction is pulling in the wheel to oppose that slide (preventing any perpendicular velocity-component) to prevent the car from slipping and sliding.
Be reminded that friction is always something that tries to prevent a motion. It will always act the exact opposite way as the motion/velocity it is trying to stop - in this case, exactly opposite to the perpendicular velocity-component of the wheel.
This introduced static friction will always be perpendicularly to the wheels direction. Any parallel component of this force would have been in the direction of the wheels rotation, and so wouldn't stop or change the motion (but just keeps the wheels rotating).
And acceleration always happens in the direction of the (net) force, $\vec F=m \vec a$.
This is why; the intuitive way.