Motion requires energy. To overcome inertia and add speed to a body, energy must be added. The energy a body has by virtue of its motion is called kinetic energy. The formula giving the amount of kinetic energy K that a body of mass m will have at a speed v is K = ½mv2. Note the occurrence of the square of the speed. That means doubling the speed does not double the energy needed but rather quadruples it. Now energy cannot be created or destroyed. So to get your car up to speed the required kinetic energy has to come from somewhere. In a standard internal combustion engine driven vehicle, this energy comes from the chemical energy that was stored in the fuel, in the gasoline or diesel.
When you want to slow down or stop the kinetic energy will be lower at the new lower speed. Since it cannot just be destroyed, this energy must go somewhere. In a normal vehicle it is converted to frictional heat in the brakes. This heat then escapes and the energy is lost forever. Every time you slow down, you are throwing away all of the chemical energy that came from the fuel. Your brakes slow you down and in the process turn gasoline into heat.
What if it were possible capture the kinetic energy that is lost as you slow down and store it for reuse later? Obviously it cannot be converted back into chemical energy in the fuel. Internal combustion engines and standard brakes have no way of capturing the kinetic energy and it must be lost as heat. This is the reason that the Golden Rule of Hypermiling is "Maintain Momentum". Every time you slow down and speed up again, you have turned some fuel into heat.
Electric engines, such as used in hybrid vehicles or NEVs can capture some energy while slowing down. This is because electric engines run off of batteries, which can both provide and store electric energy. Compare that to liquid fuel, which can put energy out but it is impossible to pump energy into. Also an electric engine run in reverse is in fact an electric generator. Depending on which direction you run it, an electric engine can draw stored energy out of a battery to provide work or it can accept outside work, generate electricity and store it in the battery. Regenerative braking basically switches the electric engine to run in reverse, in generator mode. It makes the spinning wheels and drivetrain do work on it. As the drivetrain torques against the electric engine, it slows down, thus slowing your vehicle down. The drivetrain torquing the engine allows it to generate electricity, which is then stored in the battery.
Although regenerative braking is not capable of capturing all of the kinetic energy due to inevitable losses, any percentage it does recover is available to accelerate you back up to speed without needing to put in fresh energy. Unlike normal internal combustion engines and standard braking systems, not all of your motion energy is lost when you slow. And that means you save on energy.