Synonyms for Latch on:

n.

know, comprehend, fall/fit into place, make sense of something, understand, get the idea, have a feel for something, deduce, see the light, work out. latch on (noun)
twig, catch on, get onto, fasten on, seize on, cotton on, tumble, hook on, get it.

v.

grasp (verb)
know, catch on, comprehend.

Other synonyms:

v.
comprehend.
Other relevant words:
catch on.