Understand, usually after some initial difficulty "She didn't know what her classmates were plotting but finally caught on" - get wise, get onto, tumble, latch on, cotton on, twig, get it
Become popular "This fashion caught on in Paris"
Derived forms: catches on, caught on, catching on
Type of: apprehend, change, compass, comprehend, dig, get the picture, grasp, grok [N. Amer], savvy