How has transistors change the world?

Enormously!

The predecessor to the transistor was the valve (tube in America). The valve was fragile, ran very hot, physically large and expensive to make.

The transistor (although initially expensive) is robust, efficient, tiny and practically free nowadays (a modern Intel processor contains over 500 million transistor and costs around £100 so a transistor costs 20 millionths of a penny).

So the change to the world has been the possibility of cheap "intelligent" electronics being added to many aspects of life.