If you signed a rental agreement in Switzerland, there was most likely a clause requiring you to hire an "assurance responsabilité civile privée", or Privathaftpflichtversicherung if you live in the German part. It is distinct from home content insurance, although many insurers sell them together. A lot of people sign without asking too much what exactly it covers.
Private liability covers damages that you, by accident, cause to third parties. The most classic example in Switzerland is the tap: you leave the house, the bathroom has been left open, it overflows, it floods the apartment below and the one on the next floor. Without RC, you pay for the entire arrangement of the roof, floor, walls and whatever has been damaged by the neighbors. With RC, it is paid by the insurer. It also covers if your child breaks the glass of a storefront with a ball, if you fall on someone on the street and cause them an injury, or if your dog does something to a third party, according to the terms of the policy. What it doesn't cover are damage to your own things, intentional damage, and vehicle accidents, which have their own insurance.
The cost is usually between 80 and 200 francs per year, depending on the insurer and the coverages. The reason landlords require it in the lease is exactly that: if you cause damage to the building or neighbors without having it, the problem can quickly turn into something more complicated than a breakdown.
The most frequent confusion is between CR and content insurance. The content covers your things, your furniture, your computer if there is a theft or a fire. CR covers what you can cause others. They are two different insurances although they are often contracted together and the combined premium is not much higher.