What is the function of the transistors in garage door openers?

They are part of the electrical control board circuit and typically act as switches.

Transistors are essentially switches. Before transistors electronic devices used vacuum tubes instead. The Transistors are used to tell the devices what to do when you press the button.

See How Transistors Work | HowStuffWorks

A transistor is basically a signal amplifier. When AT&T developed them it was for the purpose of replacing tubes which were considerably heavier and failed more often. Case in point, the communications systems used in WW II. Large cumbersome, heavy backpacks were replaced with much lighter and more stable radios.

Eventually, "TTL" evolved, (Transistor - Transistor - Logic) was developed. That logic morphed into what is now IC (Integrated Circuits).

A transistor receives a signal, boosts it level and feeds the supply of the IC.