I find them pretty intuitive as long as they're constructed to focus on causality: it's "what happens if I poke the system HERE?"
And then to the extent that you can model the responses as linear, or handle nonlinearities with perturbation theory, you can derive a lot from those "response to a poke at one point" results.
But in relativistic QFT it suddenly gets weird because the most mathematically elegant form of the Green's function is one that doesn't really respect causality--it involves influences "going backward in time" as well as forward. So it can be hard to see how a causal world still emerges from that.