Not a physicist, so this comment is more of a guess with the intention of someone correcting me, but I think the thing all the physicists leave out because it's probably very obvious is that when an excited nucleus returns to its ground state, it will emit radiation.
So they hit their thorium with a laser, and then instead of the laser passing through, it gets absorbed, and then they get a flash of radiation back, letting them know the thorium was excited. The delay between the laser pulse and the flash of radiation is a property of the particular thorium nucleus, and is not affected by environmental circumstances like temperature or electric/magnetic fields, so can be relied on as a very precise measurement of time.
So they hit their thorium with a laser, and then instead of the laser passing through, it gets absorbed, and then they get a flash of radiation back, letting them know the thorium was excited. The delay between the laser pulse and the flash of radiation is a property of the particular thorium nucleus, and is not affected by environmental circumstances like temperature or electric/magnetic fields, so can be relied on as a very precise measurement of time.