So basically it stems from a story about Cormac Laidir McCarthy (the builder of Blarney Castle) and his prayers to a goddess known as “Clíodhna”

Cormac had a court date somewhere in the 15th century and was quite nervous or worried so he called out to the goddess for help. His prayers where answered when Clíodhna spoke to him and told him to kiss the first stone on his way to his trial. He does this and proceeds to expertly argue his case and win.

Later it is said he had the same stone he kissed worked into his castle (the Blarney Stone) and it is said that anyone that kisses the stone will grant the kisser the gift of gab.

Side note there are other stories about the origin of the stone I believe I have featured the most “popular” one