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This is going to sound like a stupid question. It sounds stupid in my head. But here goes.

The net force on an object is equal to the mass times the acceleration, $F = ma$

When I brake on a (moving) car, the net force is negative, therefore causing the resulting acceleration to also be negative. This all makes sense, but if the acceleration of the car is negative, why does it not keep moving backward? I know cars in real life come to a stop, but I am having trouble explaining why the car does not continue to accelerate backward while the brakes are applied, with physics, so to speak.

Where is the logic incorrect?

Cars move because the wheels are spinning in a certain direction. Brakes work by making the wheels not spin, not by making them spin in the opposite direction.

If instead of slamming the brakes you "brake" a car by having some other kind of force pushing it backwards, like a super huge fan in front of it, then yes, it might begin moving backwards.

A notable property of frictional forces is that they resist motion (as opposed to other types of forces, which might resist displacement, for example, which is how a spring behaves). As a result, the brakes on your car slow down the motion of your wheels that produce forward movement of your car—but they also slow down motion that would produce reverse movement.

If instead you used another type of force-applying mechanicam to slow down your car (e.g., a giant spring), then your car would slow down, then stop, and then start moving backwards.