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To take a statement with ‘a grain of salt’ or ‘a pinch of salt’ means to accept it but to maintain a degree of skepticism about its truth.
Origin

take with a pinch of saltThe idea comes from the fact that food is more easily swallowed if taken with a small amount of salt. Pliny the Elder translated an ancient antidote for poison with the words ‘be taken fasting, plus a grain of salt’.

Pliny’s Naturalis Historia, 77 A.D. translates thus:

After the defeat of that mighty monarch, Mithridates, Gnaeus Pompeius found in his private cabinet a recipe for an antidote in his own handwriting; it was to the following effect: Take two dried walnuts, two figs, and twenty leaves of rue; pound them all together, with the addition of a grain of salt; if a person takes this mixture fasting, he will be proof against all poisons for that day.

The suggestion is that injurious effects can be moderated by the taking of a grain of salt.

The figurative meaning, i.e. that truth may require moderation by the notional application of ‘a grain of salt’, didn’t enter the language until much later, no doubt influenced by classical scholars’ study of Ancient Greek texts like the works of Pliny. The phrase has been in use in English since the 17th century; for example, John Trapp’s Commentary on the Old and New Testaments, 1647:

“This is to be taken with a grain of salt.”

The ‘pinch of salt’ variant is more recent. The earliest printed citation that I can find for it is F. R. Cowell’s Cicero & the Roman Republic, 1948:

“A more critical spirit slowly developed, so that Cicero and his friends took more than the proverbial pinch of salt before swallowing everything written by these earlier authors.”