It's believed to have been a part of black American slang for decades, possibly from the term "high-capping" which essentially means bragging. Rapper E-40 used the term in his 1995 song "Da Bumble".
Turns out cappin' is in opposition to no cap (a video game term for an accurate sniper shot)... So where no cap came to mean accuracy, "cappin'" came to mean inaccuracy or deception/lying.