"Chick chick chick" is a fairly close approximation of the "food call" a broody hen gives when she's found something for her chicks to eat, so it's not surprising that a lot of adult birds respond to that too! The rooster food call is similar, but a lot deeper.
Our birds know the blue "kitchen bucket" very well--it sits in the corner of the kitchen and collects scraps of fruit, veggies, cheese, and bread. If we go out there with that bucket, they all come at the run. I have a much larger red bucket that I use as a mobile toolbox for projects, and they come running for that too, but they seem more curious to see what's in it than expectant of food.
Regarding chicken cognition: they're a lot smarter than rumor gives them credit for being, actually. They have a very strong prey-animal flight reflex, and when startled they act like complete idiots, which does nothing for the reputation of their intelligence. But if you work with them when they're not agitated, they do quite well; they're no parrots, but they ain't dumdums either. Ours do have some word-association ability--although it's heavily dependent on tone of voice. Our rooster in a wheelchair has learned that "Hup-up" means I want him to stand so that I can lift him out of his sling, and most of the birds that get handled a lot know that "Stop" means stop flapping and panicking.
For the most part I think they learn through trial and error to associate meaning with certain human "calls" that they hear a lot, just as chickens have their own calls. And I think they categorize human calls the same way as chicken calls: this one means food, this one is a greeting, this one means they're getting mad, this one is alarm, etc. Certainly chickens have a fairly varied vocabulary of their own, even when very young.