Rivers and streams wash tiny amounts of salt out of the rocks and soil. This of course all fetches up in the sea.
Water from the sea evaporates, becomes clouds, and falls on the land -- it goes in a cycle. Salt does not evaporate, so it's a one-way trip for every tiny bit of salt in the rivers.
Eventually that builds up...
Salt can only get out of the sea by part of the sea getting cut off as a salt lagoon and dried up. This has happened on many occasions in geological history, and if later buried by mud and sand, it gives us the beds of rock salt which occur in a few places.
However, the amount of rock salt is tiny compared with the whole sea, so that remains salty, and gradually becomes more so.
The blood of vertebrates is less salty than the sea now is, probably because when they (and thus we) left it, it was then not so salty.