The salt is specifically to invalidate pre-generated rainbow tables, and doesn’t need to be kept private. It only needs to be unique.
The attacker generates rainbow tables by running common passwords through known hashing algorithms. So I run “password1” through a bunch of different algorithms, and save the results of each. Notably, generating decently large rainbow tables takes a lot of time, processing power, and storage space. Because you don’t just use common passwords; You’re basically running a brute force/dictionary attack on your own computer’s hashing algorithm.
Now if a database is unsalted, I can search for matching results against my rainbow table. When I see a match, it tells me both which users had that password and which hashing algorithm they were using. So now I can narrow down my focus to only using that algorithm.
But if a database is salted, all of my pre-generated tables are useless. Even if someone used “password”, it won’t match my rainbow tables because the hash was actually fed “password{hash}” instead. And even if multiple users used “password”, each salt is unique, so I don’t see a bunch of repeated hashes (which would point to those accounts using the same password). I would now need to generate all new tables with the salts I stole in order for my rainbow tables to be usable again. And even then, I’d need to repeat that table generation for every user.