What is the physical (conceptual) importance of corner frequencies?

They are the transition frequencies, with values depending on the components of the network, between different regimes of frequency response. I'm not sure there's anything deeper than that, in general, although for specific networks you can think of them physically. For example, for the low-pass filter, the corner frequency is $\omega_c = 1/(RC)$, which is 1 over the time constant of the network; for frequencies larger than that, capacitor never fully charges up so you never develop the full voltage on the output (so the sinusoidal signal is attenuated); for frequencies smaller than that, you charge and discharge the capacitor up and down all the way in each cycle and output follows input (see also some of the answers in FAQs 8 and 9).