In order to compete on my horse here in Switzerland I need a license (maybe not on all competitions, but the bigger ones).

Does anyone know how I do to get this?

French/German

as indicated in the link above, you need to have a Brevet for basic competitions (and remember to pay the 'validation fee' every year that you want to compete) and a Lizenz for higher competitions. Both need to be taken at accredited stables (see list on link), you can use your own horse though.
Also, your horse needs to be officially registered at the FNCH and you need to pay every year (as well) to keep that registration active.
This is to my best knowledge, as I have been thinking about competing as well and my riding instructor gave me these hints.
Otherwise you can call the FNCH, they are supposedly really nice and friendly, to make sure you get all correct registrations.