Is someone yapping on and on and you only have the vaguest idea of what they're talking about? Then you understood just an inkling — a glimmer, a fraction — of what they were saying.
Inkling can also mean a sly suggestion or faint implication. If someone drops a hint you're not wanted they've given you an inkling you're not wanted. The word comes from the medieval English word inclen, which suitably enough means "to utter in an undertone." In other words, what's really being said is in between the lines of what's actually being said on the surface. By now you've probably got the inkling that inklings can be sneaky things.