A Luxembourg chauffeur service is a private car booked in advance at a price agreed before the trip, driven by a professional chauffeur who works for an established operator. It is not a taxi. Luxembourg licenses its taxis nationally through the Ministry of Mobility and Public Works (Ministère de la Mobilité et des Travaux publics) under the law of 5 July 2016, while pre-booked cars with a driver, known as location de voiture avec chauffeur or VLC, currently need neither a taxi licence nor a taxi driver card. What you book is the car, the chauffeur and the euro amount, whether that is a Luxembourg airport transfer, an hourly car service in Luxembourg City or a black car to a Kirchberg meeting.
The case for it is mostly about the border. Close to half the jobs in the country are held by people who live in France, Germany or Belgium, and they arrive by road at the same hour every morning, so the A3 and A31 from Thionville carry up to 4,000 vehicles an hour at the frontier. A reserved Luxembourg car service does not change its price when that traffic stalls. The figure you accepted already contains the 3 percent VAT on passenger transport and any airport kerb charge for your pickup. For the honest comparison with a metered ride or the free tram, see the Luxembourg taxi alternative.
The same booking reaches well beyond the city. Frankfurt Airport to Luxembourg and Brussels Airport to Luxembourg cover long-haul arrivals the local airport does not serve, Hahn to Luxembourg picks up low-cost flights in the Hunsrück, and Luxembourg to Metz or Luxembourg to Trier handle the daily cross-border trips. Institutions, banks and fund houses moving staff and visitors through Kirchberg every week can add central billing and reporting through corporate travel in Luxembourg.