I understand the basic definitions. Locality means Alice's measurements do not affect Bob's and system and that no-signalling means a party can't send information faster than light. I also know that locality is defined at the ontological level but no-signalling is at the operational level. What I do not understand is why it is so. Can this be understood with physical intuition?
I think the best way to understand this is by showing that one can violate (ontological) locality while respecting (operational) no-signalling.
Take the case of Bohmian mechanics. In it the result of Alice's measurement is deterministic, and it will depend Bob's choice of measurement, so it is clearly nonlocal. Nevertheless, the "quantum equilibrium" axiom postulates that the hidden variables are distributed in such a way that they reproduce the statistics of quantum mechanics, so you can't use it to signal, as quantum mechanics cannot. If you violate the "quantum equilibrium" axiom, simply by knowing the value of a hidden variable before the measurement, you can use this knowledge to signal faster than light.
I hope this makes it clear that "no-signalling" is really about what you can do with the theory, it's an intrinsically operational definition. As for "locality", it's a bit harder to say because there are several different definitions of locality, but usually it is about what is actually going on behind the scenes, independently of what you can do with it, i.e. it's usually an ontological definition.
No signaling means can't be used for communication. Can't be used to move messages from a sender to a receiver.
Local means doesn't require communication to implement.
Some processes, such as phenomena violating bell inequalities, don't enable communication but also can't be emulated classically without communication. They take without giving back. They are not local but not in a useful way that would allow you to embed messages into that non locality.