In cavity optomechanics, what does it mean that a cavity mode is driven by a classical field?

It means that one or multiple channel modes (an external mode in the Jaynes-Cummings Model of CQED) of the cavity are driven by a coherent state. Here "driven by a coherent state" is to be understood within the Input-Output formalism: The input operator is in a coherent state.

It turns out that for a coherent state the quantum stochastic equations of motion can be replaced by an effective Hamiltonian that is equivalent to a classical driving field Hamiltonian.