The binary system (1s and 0s) is the basis of computing. The transistor allows you to control the flow of current, on for 1 and off for 0.

Other electronic devices can do it too, for example vacuum valves.

But by construction, a transistor can be miniaturised so that millions of them fit on a single silicon chip. You couldn't do that with valves. They also require far less power to operate. That's the real key to their importance in computing: less space, less power, lower cost and higher capacity.