It's quite like 'not completely'. The salt indicates something that is added to a dish, and therefore the food no longer has its original taste.
The same can be true of a what someone says. You can believe them to a degree, but you should understand that there may be other things to consider. What someone says is not always the full truth, or they may not have all the facts about a situation, etc. Therefore you take what they say 'with a pinch of salt'.
It indicates caution.
I must add that I am not a teacher, so others may well be able to give you a better (more formal) description.
