Because agriculture and manufacturing can be, and have been, largely automated in those economies.
But you still need to employ people, and services cannot be as automated as the other two: a human guide can answer questions an audioguide cannot. A waiter or waitress can give you advice and do small talk.
Also a developed economy has historically meant more free time. Time which is now put toward leisure, which means you need more people in service to accomodate for people using those services.
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