I understand that approximate and detail coefficient represent the different signal bands. But what do the values mean and how are they used?
DWTs turn a signal into a list of coefficients, whose combined sizes are equal to or greater than the signal length. Coefficients are grouped in packs called subbands. Each subband gathers coefficients resulting from some band-pass filtering, followed by a subsampling.
They are not exactly narrow-band signals, but projections onto embedded subspaces. The "narrowband" signal for each subband could be recovered by inverse transforming those coefficients only.