Veasey is a lifelong resident of Fort Worth, Texas, having been born there on January 3, 1971. His parents, Joseph and Connie Veasey, divorced when he was ten, prompting Marc, his brother Ryan, and his mother to move in with his maternal grandmother. Prior to this move, the family had been relocating frequently from rental house to rental house in Fort Worth. Upon graduating from high school, Veasey attended Texas Wesleyan University in Fort Worth where he earned a B.S. in Mass Communications in 1995.

After graduating from Texas Wesleyan University, Veasey wrote scripts for advertising agencies and worked part-time as a substitute teacher. His first foray into politics began after he volunteered in the summer of 1999 to work for U.S. Representative Michael Frost, who hired Veasey as a field representative. Veasey continued to work for the congressman for five years.

In 2004, Veasey challenged incumbent State Representative Glenn Lewis of the 95th Legislative District in the Democratic Party primary and won. In the general election, Veasey had no Republican challenger and ran unopposed. He was consecutively re-elected three times. During his time in the state house, his most notable achievement was the authoring of HB 62 that honored Tim Cole, an African American student who had been wrongly convicted of rape in 1985.