I can imagine that ever since there were more than two groups of people it would have been normal to send someone from group A to talk to group B. But in modern times everyone has ambassadors and embassies in multiple countries staffed year round. How did this begin?
Hey there, I answered a similar question in greater detail over in this thread the other day.
To directly address your question; permanent ambassadors first start cropping up in northern Italy in the 15th century. Relations between the powerful city states in the region were so dynamic that having a permanent ambassador became a matter of necessity. At the same time, interstate relations in the rest of Europe were becoming more widespread and complex, which created a similar need for a more complex and formalized diplomatic system. The late medieval and early modern periods sparked a lot of diplomatic innovations, with formal, permanent ambassadors being just one of them.
Getting back on topic, the need for a permanent, formal ambassador was not yet so great in the rest of Europe as it was in Italy. This led to a bit of an awkward transition period. In the middle ages, diplomats would be sent abroad to do one specific task before returning home. Wanting to gradually establish their ambassadors as more permanent fixtures in foreign courts, ambassadors were continuously sent a list of menial tasks to do as a justification for their continued presence. Thomas Spinelly, Englands proto-ambassador to the low countries in the early 16th century, served during this awkward transition period. Historian Betty Behrens wrote a great article that discusses Spinelly's service as part of a legation in limbo if you'd like to read more on how exactly that worked, linked below. By the 17th century, the idea of a permanent ambassador and permanent diplomatic staff running foreign embassies and consulates was well established.
Sources: Anderson, M. S., The Rise of Modern Diplomacy, 1450 - 1919 (London, 1993).
Black, Jeremy, A History of Diplomacy (London, 2010).
Mattingly, Garrett, Renaissance Diplomacy (Boston, 1955).
Mowat, R. B., A History of European Diplomacy, 1451–1789 (London, 1928).
Behrens, Betty. "The Office of the English Resident Ambassador: Its Evolution as Illustrated by the Career of Sir Thomas Spinelly, 1509-22." Transactions of the Royal Historical Society 16 (1933): 161-95. doi:10.2307/3678668.