The function of backing plates is to hold the braking system together so it can work properly. With drum brake systems, each wheel has two curved brake shoes mounted on the stationary backing plate. The brake shoes are shaped like the letter C with friction material on the outside curve. They're attached to either side of the brake plate. This assembly is, in turn, mounted inside a cast-iron drum that rotates along with the wheel. Springs hold the brake shoes in place so they don't contact the drum when the brakes aren't active.
When the driver pushes down on the brake pedal, this places pressure on the hydraulic fluid in the brake lines. At the top of the backing plate is a wheel cylinder containing two pistons, one at either end. The pressure from the brake fluid enters the wheel cylinder and forces the spring-loaded pistons to move outward from the wheel cylinder, pushing the brake shoes against the inside of the brake drum.
The friction material on the outside edge of the brake shoes coming into contact with the inner surface of the brake drum is what stops the car. But without the stability and function of brake plates keeping all of these brake parts in place and taking the brunt of the braking force, none of it would work.
If you want some more information about backing plates and the other forces that bring your car to a halt, stop by the next page.