To simplify it:
You have a radioactive element (americium) in the detector on one side of a chamber. On the other side of a chamber is a detector. The radioactive source emits particles every so often. If the detector on the other side receives these particles, all good, nothing happens.
If smoke gets into the chamber, it blocks the particles coming out of the americium. So the detector doesn’t detect any particles, and it activates the alarm.
The link is just giving a few details on how the detector actually does that.
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