It's a metaphor. The other person is looking serious and is obviously using mental effort to try to understand something.

The term "gears turning" refers to someone thinking very seriously about something. Imagine your brain is a clock and the "gears" inside of it are working so hard that you can hear them moving. Example: "I know you are trying really hard to figure out the answer to this difficult mathematical equation; you are thinking so hard I can hear your gears turning"