The key is to recognize what the unemployment rate is actually measuring. It's not the proportion of people without jobs; it's the proportion of people seeking jobs---people who don't have jobs and want to have jobs.
When someone becomes a full-time student, or retires, or becomes disabled, or even simply gets so discouraged they give up on finding work, they are counted as "not in the labor force"---so they do not factor into the unemployment rate.
The rate of people who don't have jobs for any reason is the nonemployment rate, and it's always vastly higher---in most countries as high as 40% or even 50%. But while unemployment is a macroeconomic problem, nonemployment is not; having a lot of students or retirees isn't necessarily a bad thing.
This is why adding up the number of unemployed people plus employed people does not give you the country's whole population; you have to add in everyone who is not in the labor force for one reason or another.
Thus, if a lot of unemployed people leave the labor force by becoming students, retiring, or just giving up, the unemployment rate can fall, even though nobody actually got a job.
There's another way the unemployment rate can fall, which is if the unemployed people emigrate to other countries. Then they wouldn't even be counted in the nonemployment rate, because they're not in that country's population anymore. This rarely happens in the US (a lot more people migrate into the US than out of the US), but it does happen in many other countries, ranging from Lebanon to Nicaragua to Uganda.
There are some newer measures of "broad unemployment" which include "discouraged workers" (people who give up on finding work) and "part-time employed for economic reasons" (people who'd like to work full-time but can only find part-time jobs); these measures may be better for accessing macroeconomic health, but they're relatively recent and we don't have good data on them far back. They also still wouldn't show a rise in unemployment if people became students or left the country.
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