As you know, sea is salty and not suitable for humans to drink, but one can drink the fresh cold water from a clean river without any harm. This does not mean that rivers are not salty.
As water falls from the clouds, they go through the ground picking up minerals and salts that can be easily dissolved into water. River water has only a small amount of salinity and that's why we do not taste it. It is the same as if you pour a pinch of salt into a glass of water...
This water eventually pours into the sea. Salts and minerals accumulate, but the water mass evaporates again to go though the same water cycle, over and over again for millions of years. Salts cannot be evaporated though so they accumulate over time.
This does not mean that the sea gets saltier every day though. Sea salinity has reached a steady state. The amount of new salts introduced into the sea equals to the amount of salts being removed by various processes including chemical reactions and sedimentation.
Bonus knowledge: What's wrong with Great Salt Lake and the Dead Sea and they are that much salty? Well, lakes are usually temporary buffers of river water. There is one or more entries from where water enters, and one or more exits from where the water exits the lake. For Great Salt Lake and the Dead Sea, there are only entries. So, water along with minerals and salts enter the lake, but there is no exit for the water to carry the salts away. Eventually water is evaporated leaving large amounts of salts behind. And since these lakes are much smaller than the ocean, they are not capable to reach a "low" salinity steady state.