How does a repeater convert light from a fibre-optic cable into binary?
How does a repeater convert light from a fibre-optic cable into
binary?

The received light is converted to an electrical signal using (typically) a photodiode: -

It converts incident light power into a current. The current is (usually) converted to a voltage using a TIA (transimpedance amplifier): -

And the voltage can be converted into a series of logic bits using a voltage comparator: -

That's the simple story.
The more complex story involves automatic gain circuits in case the light level changes over time and distance (usually the case). The more complex story can involve a tweaking of the hardware based on what data is received i.e. if error rate appears to be a little too high, there can be feedback to the hardware that tweaks things this way or that way in order to reduce the error rate.