Nervous; high-strung. A wart is, of course, a bothersome bump on the skin which is a harmless but disfiguring scar, so a worry wart is like an unnecessary node of  alarm that serves no useful purpose.  (Compare to Nervous Nellie or Hand-wringing.)

“Ever since I  earned my driver’s license, my mom has become a worry-wart every time I drive the car.”