How can a flywheel make engine run smoothly? In the video Moment of Inertia Demo, a man states that moment of inertia will be higher if the masses are away from the axis of rotation. 
The engine take more power to spin the flywheel, because of its huge mass. Efficiency of the engine drops very low. I can only see more burden than smoothness. 
This is the sample picture of an exposed flywheel.

How does a flywheel make an engine run smoothly?
 A: A flywheel requires energy to spin up, but can also return that energy when needed. In a one piston engine, some energy from the combustion is used to accelerate the flywheel. However, in the compression phase, it returns some of its energy to the piston and helps retaining the rotational speed. Because the speed doesn't drop as much, the next combustion doesn't have to accelerate the flywheel that much. The exchange of energy from and to the flywheel cancel each other out, so it doesn't impact efficiency, but through its rotational inertia it helps keeping the rotational velocity more constant.
A: A steam or internal combustion engine delivers energy in strokes as each cylinder in turn is filled with steam or burns its portion of fuel. But the load driven by the engine should usually rather be rotating at a steady speed than moving in strokes.
The flywheel does this job. It consumes the energy from cylinders as they release it -- in strokes, by speeding up -- and gives it to the driven mechanisms the rest of the time -- by gradually slowing down. The faster the flywheel rotates, the steadier its speed becomes because its kinetic energy is $J\omega^2/2$ and the portions added by cylinders (which are (roughly) constant) become lesser fractions of it.
A side benefit of this is that the control mechanic of engine's own cylinders can be driven by the flywheel, too. A drawback of this is that a separate "starter" facility is now required to drive it while the flywheel is not yet moving.
A: 
By adding flywheel makes engine to take more power to spin the flywheel because of its huge mass. Efficiency of the engine drops very low. I can only see more burden than smoothness.

The burden is temporary in order to get the flywheel going. Once it gets going the stored  rotational kinetic energy reduces the energy required by the engine. Refer to the figures below.
A flywheel is designed to efficiently store rotational kinetic energy. It resists changes in rpm by virtue of its rotational  moment of inertia. The stored energy is proportional to the square of its rpm and proportional to its moment of inertia.
The stored rotational kinetic energy in the flywheel helps to keep it rotating thereby reducing the combustion energy required from the engine. The power stroke of the piston/cylinder, which follows figure 5 when the flywheel returns to its position in figure 1 (to start the compression stroke), is only needed to make up for the relatively small loss of rotational kinetic energy when the flywheel does positive work (compression) on the gas during figures 1-5. 
Hope this helps. 

