A person with the stolen database tables has only 2 fields per user.
password hash salt
c32528b3e9191a8c7471252c6de289939a1e6f01 cGFzc3dvcmQ
In the most basic terms, the way I understand it, all what salting does is make the password more longer.
The original password was 'password'. But I appended to the password before computing the hash the salt cGFzc3dvcmQ to make passwordcGFzc3dvcmQ. What does that do? Well, it just makes it so if you happen to have a rainbow table of common passwords and their hashes,
password hash
password 5baa61e4c9b93f3f0682250b6cf8331b7ee68fd8
where you can see at a glance that SHA('password')=5baa61e4c9b93f3f0682250b6cf8331b7ee68fd8, looking up the original password from the hash is all too simple. We want the cracker to have to do some work for it.
So we go "Dyoh! We either prepended or appended THIS RANDOM STRING RIGHT HERE to the original password! Now your hash tables of common passwords and their hashes are useless"
Which they are. Because we complexified each password by the random string, basically "making the user's passwords better", and completely screwing up the hashes so they are completely uncommon now.
So salting makes it so each hash has to be cracked individually, because to do it, if the cracker has the salts, the cracker has to append/prepend the salt to each common password he tries. 2 different users could use the simple password password, but cracking that will be 2 entirely separate jobs for the cracker, because of the salt. So cracking all the passwords will take longer.
The article here says that because of the compute power available today, however, using salts is trivial to crack.