The two-word explanation: founder's advantage.
The modern chocolate industry developed in the 19th century. The first key step was Dutch cocoa, developed in the Netherlands in the first 1/4 of the 19th century. Part 1 was the extraction of the fat (cocoa butter) to produce cocoa powder using a hydraulic press (compared with the older process of boiling and skimming the fat). Part 2 was an alkali treatment of the cocoa powder to reduce bitterness.
The second key step was the development of block chocolate, made with cocoa powder, sugar, and cocoa butter. The inventor was the British chocolate firm Fry and Sons. They were already industrially making chocolate products, and were set to become a big manufacturer of block chocolate. Fry's early major competitor was Cadbury. Rowntree entered the market later, with success. All three of these British manufacturers owed a lot to their adopted of the Dutch process.
The third key step was milk chocolate. Henri Nestlé, a German migrant to Switzerland, was working on infant formula. As part of this, he had developed powdered milk. Nestlé's neighbour, chocolatier Daniel Peter, latched onto this and developed milk chocolate. Together, they formed the Nestlé Company, which still dominates drinking chocolate and powdered milk markets. What was the impact of milk chocolate? Huge! Milk chocolate is the single biggest part of the chocolate confectionery marker (followed by filled chocolates, and dark chocolate). Milk chocolate was adopted by other manufacturers (the British firms noted above, and American companies such as Hershey's).
The fourth key step was conching, mechanical treatment of the chocolate to give improved texture and flavour. This was developed by Lindt.
Milk chocolate and conching made the Swiss brands (in particular, Nestlé and Lindt) big. The big 19th century brands are still big in the world chocolate market, although some of the companies now have different owners (Fry's was bought by Cadbury in the early 20th century, and Cadbury is now owned by the US multinational Mondelez International (also the owner of Toblerone)).
The big brands dominate the market, but the smaller luxury makers (combined) have a sigificant share of the market. Many of them are non-Swiss (e.g., Ferrero Rocher of Italian, Godiva of Belgium, and Valrhona of France). As a generalisation, the big brands are more important in the block chocolate market, and the luxury brands are known for their filled chocolates.
Given the early British advantage in developing block chocolate, if the British firms had proceeded to develop milk chocolate and conching, it's unlikely that Swiss brands would have gotten any significant part of the market. Nestlé's decision to move from Frankfurt to Switzerland had major consequences - he was the butterfly whose flapping wings made a hurricane in the global chocolate market.
There's always more to be said, but I wrote up an answer to a similar question some time ago:
When chocolate was first imported from the Americas, it was little like the bars you're used to today - for one thing, it was typically consumed by North/Central Americans as a bitter, cold drink. The Spanish began to add spices and sugar in order to make it more palatable, and heated it up; they also routinely ground the cacao beans ahead of time and formed them into solid blocks for transport across the Atlantic, which could be broken up and dissolved in hot water with sugar to make hot chocolate. In the early nineteenth century, Coenraad Van Houten invented a press for fully splitting the cocoa butter and cocoa mass, as well as the Dutch process, which made the powdered cocoa mass taste sweeter and mix better with water - and by combining the Dutch-process cocoa powder with the cocoa butter, it was possible to make an edible solid chocolate! (Pieces of edible chocolate were first made by the British company Fry & Sons, a rival to Cadbury, and in 1847 they had developed a way of mixing cocoa powder, melted cocoa butter, and sugar into a paste that could be shaped into chocolate bars.)
For the most part, European hot chocolate had been a luxury product, consumed by the elite, but by the end of the eighteenth century it had trickled down to the masses. It was particularly popular in Italy, where it was even added to normal mealtime foods, and people who were interested in learning the trade often went to learn from Italian confectioners' expertise - people like François-Louis Cailler, the creator of the first chocolate factory in Switzerland in 1819. He was followed in the nascent Swiss chocolate industry by Philippe Suchard and Charles-Amédée Kohler within the next decade. While Kohler's pretty important for inventing the idea of a chocolate bar with nuts in it, the real player here is Cailler's son-in-law, Daniel Peter, a generation later. Peter had come up with the idea of adding milk to Fry & Son's paste mixture to make a creamier bar on his own, but simply couldn't get it to work: it became mildewy. But Henri Nestlé, an industrialist who had developed powdered milk as a potential alternative to wet-nurses, was living and working in the same town, and the two were able to put their products and knowledge together to combine powdered cacao and powdered milk, removing the water that caused the mildew from the equation in 1875. Milk chocolate!
Rodolphe Lindt, a former apprentice under Kohler, worked off of these earlier developments to create the process known as "conching" at the end of the nineteenth century. Conching is basically running all of the ingredients - cocoa powder, cocoa butter, sugar, etc. - in a mixer continuously for several days, which makes sure that the entire thing is as homogeneous as possible, the cocoa butter perfectly distributed and the texture smoother. Instead of a paste, the result is a liquid that can be poured into molds, too, which has implications for speed of production and the product's look.
It's after all of this had been done that Switzerland came to be at the forefront of the chocolate industry. At the very beginning of the twentieth century, many of these historic firms were consolidating, combining their wealth and factories to produce more and more and more chocolate - so much that Switzerland's population couldn't buy it all. In response, the companies began to market their products to the rest of the world, and concentrated on improving the quality of chocolate to outshine competitors from other countries rather than innovating in other ways (coming up with new textures or fillings, etc.).
Edit: It's worth mentioning that Belgium is now considered to be a source of high-quality chocolate, actually - but that's a much more relatively recent reputation. Belgium only began to export more chocolate than it imported in the 1960s, when a deliberate effort was made by the government to promote "Belgian chocolate" as a kind of brand.