From the French chauffer, meaning ' to heat'.
The first people to be known as chauffeurs were a particularly gruesome class of thief. They would brake into people's houses, and torture them by burning their feet in the fireplace until they told where their valuables were hidden.
Later it was applied to stokers of steam engines, because they kept the engines hot. When the first steam driven automobiles came out, they too had chauffeurs. Steam driven cars didn't last long, but when the first gas driven cars came out, the drivers were known jokingly as chauffeurs. Of course, these were not hired help, but the rich owners of the automobiles (we would simply call them motorists).
Today rich people hire other people to drive their cars for them, and rich people like fancy French words like 'chauffeur' better than English ones like 'driver'. So since 1902 a chauffeur has referred to the hired drivers.
By 1917 chauffeur had started to be used as a verb, meaning 'to drive someone about'. And that is where the situation stands today.