How does it all work?

That is quite a complicated question. To fully understand how a cable car operates, you would probably have to go to the Cable Car museum, but here is a little excerpt"

'What's Under the Street?'
"... [There] are eight large wheels, called 'sheaves,' which change cable direction going into and out of the powerhouse. There are four cable lines in San Francisco, each a closed loop. All four cable lines enter and leave the powerhouse at this location."

"Hundreds of small sheaves are under the streets where the cable cars run. They keep the cable aligned and make curves and crossings possible. These smaller sheaves are referred to as pulleys."

'What's Underfoot?'
"The cables pass directly under your feet on their way to the motors, gears and more sheaves which keep the cables constantly moving. These motors, gears and sheaves are called the 'window machinery.'"

The car is connected to the cable by what is called "The Grip" which is described as essentially "a giant pair of pliers" that grabs and slides onto the cable at varying levels of strength.

From my journey to the Cable Car Museum.