Why the chain of bicycle is to be greased?

Actually they don't. Today you can get belt driven bikes, or shaft driven bikes. And back in the days there used to be direct-drive bikes with the pedals solidly bolted to the wheel axle. There's also been a treadle-driven bicycle.

But direct-drive makes it awkward to find a comfortable riding position, and to get any speed you end up with really big drive wheels. Treadle-drive needs more moving parts interacting. Transmission shaft bikes tend to experience"wind-up" when being pedalled hard, and quite a bit of power is lost in the type of gearing that is needed in the power transfer train. Belt drive can be quite good, but as the belts can't be opened or cut to size, these bikes usually need special frame designs.

A chain is easier to work with than a belt, has simpler mechanics than a treadle, less mechanical losses than a shaft, and better speed/fit options than a direct drive, so it became the most common design.