Options are essentially like insurance. If you buy insurance you pay a premium. If you sell it you earn the premium (provide insurance).
A put option is guaranteeing you a certain (strike) price even if the market value drops below the strike. In that case, the owner of a put (who is long or bought the option) paid for the right to sell at a predetermined price.
A call option allows the owner (who is long or bought it) to buy at a certain (strike) price. So even if the prevailing market price is above that value you can buy it cheaper.
In both cases the option buyer has bought a right that they can use if the market moves. The option seller only earns the premium and needs to honour the contract if it is profitable to exercise for the owner. Similar to insurance companies again, where the best scenario for the seller of insurance is that there will not be a claim (and they keep the premium).
@Regio's answer is wrong. If you are short a put, you sold a put. You cannot hope for any future profit as you are selling the right to sell to someone.