Geneva is the second biggest city in Switzerland after Zurich. Given its world-class nature and the presence of big NGOs and the predominance of finance and banking sector, it does attract the frequent fliers. Geneva has many things that interest the rich of this world... go shopping in the city center streets and see all the clockwork and jewelry shops!
The second point is its central location in the Alps. People don't land in GVA just to stay in the city: many transfer to travel further inland. It is simply the closest airport with regular traffic to most of the renowned ski resorts.
Many transfer shuttles wait behind the arrivals door to bring passengers to all the famous ski places in Switzerland, and also in neighboring France. The border is right behind the runway, opposite the terminal, and the airport has a landside sector in France with maybe the only landside customs checkpoint I have ever seen in any airport. Chamonix is easily reached by road in 1 hour. When landing in GVA, you can quickly walk to the station, less than 5 minutes by foot separate baggage belts and train tracks. From there leaves a half-hourly train to Brig serving all of Valais, the canton where the biggest resorts are located like Verbier, Crans-Montana, Zermatt, Saas-Fee... Most require an additional transfer but transportation is very good after all, including availability of cogwheel trains linking the main railroad and the resorts.
There is another airport closer to the Swiss slopes, which is Sion (SIR). It has very little scheduled traffic but a runway capable of landing 737s. It used to survive due to the presence of the army who closed its base recently; this airport sees some seasonal ski traffic with weekly flights on Saturdays. Swiss Air tried to see if the route is viable, and a new airline, Powdair, will use it as it is centered on ski destinations. Approach is difficult due to the mountains and the hospital that must be avoided during descent. Smaller aircraft like Embraer are used, which opened the possibility of operating out of London City. The terminal is a small building with no jet bridges and just 4 check-in counters. Sion is very close to the big resorts but Geneva has a huge advantage with the traffic it can handle.
What's more, easyJet has a subsidiary, easyJet Switzerland, which uses GVA as the primary hub. Look for the Swiss flag on planes and the "registration number" beginning by HB- which indicates Switzerland.
Your question is totally understandable as the airport is damn small for a "regular" airport, at least on the landside. It really gets crowded during peak times. Even more strange but true: for the same reasons, you can fly nonstop to Dubai, Abu Dhabi, even New York City with both Swiss and United serving the route!
Concerning Zurich which is the big hub of Switzerland, it likely has bigger costs for airlines. Zurich is located on the plateau, further away from the mountains, making it less interesting for ski destinations. Maybe the ski resorts of Graubunden, Interlaken, Lucerne area and central/eastern Switzerland are better served from ZRH than GVA. Even the resorts of upper Valais like Zermatt can still be conveniently within reach by train from the airport since the Lotschberg base tunnel opened in 2007 and shortened train travel times.
Other big airports close to the Alps are Lyon St Exupéry, France (LYS) and Milan Malpensa, Italy (MXP). Both are further, Lyon has no direct rail link, maybe busses. Milan can be an advantage on the air fares; many flights are cheaper from Milan than from Switzerland to identical destinations.