Silver is used for trading with traders, and as a construction material. A decent amount of starter silver is among the wreckage of the colonists pods at the beginning of a new embark. More silver can be found through mining, trading, and defeating raiders.
Each trader has a limited stock you can buy from, and with silver, the player can use it to buy slaves, resources, and weapons. Additionally, traders will buy resources relevant to their specialization, such as pirate merchants buying slaves, and bulk merchants buying stone, steel, or gold.
Colonists can suffer from a "High Expectations" mood debuff if your colony becomes too rich. Make sure you have plenty of mood-boosting effects to buffer against this before you start accumulating wealth.