How can they show us the passage of time? And how is it so exact?
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It depends on the clock. But all of them rely on some repeating process that takes a known amount of time.
In a pendulum clock, the time it takes for the pendulum to swing is known. Under some simplifying assumptions, the time for a pendulum to swing is dependent only on its length, so there’s a specific length that corresponds to a swing time of 1 second. You hook that pendulum up to a gear, then set that gear up with other gears attached to the hour, minute, and (if desired) second hands with the appropriate ratios.
Electronic timekeeping involves special circuits that produce a periodic signal called, appropriately enough, a [clock signal](https://en.wikipedia.org/wiki/Clock_signal).
The most accurate clocks are atomic clocks, which work by measuring specific vibrations in specific types of atom. Specifically, they work by finding a resonant frequency in specific atoms and tuning light of exactly that frequency. Since the frequency and energy of light have a nice relationship, and since you can precisely measure how much energy you’re putting in to a signal, you can measure time extremely accurately (with an error of about 1 part in a billion trillion) this way.
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