Americium-231 is used because of its particular half-life. The rate is not too fast so as to be harmful to the humans in the house, but not too slow so the smoke detector doesn’t work.
Smoke detectors work through the process of nuclear ionization. As Americium-231 decays, the particles and energy released from that decay ionize the oxygen present in the air. This allows the air to conduct a charge to the detector, and keep the alarm from ringing. However, if smoke is present, the alarm will go off because smoke cannot be ionized through nuclear decay. As the concentration of smoke builds up, the voltage drops at the detector and eventually triggers the alert.