There are a few ways to quantify the dissolution. The older style, a typical brand is Distek, requires you to pull a sample at the correct time point using a syringe with a cannula. Methods vary, but there is a certain depth specified in the individual methods, defined by the protocols you manufacturer works under -- either USP (United States pharmacopeia, for sales in the US), there are also EP, JP and maybe there is still BP, and there may be more.
At any rate, once you've sampled from the six (or more depending on the experiment) chambers, you can run the samples in an HPLC (perhaps even a GC, if the active is volatile enough), or you can analyze by UV. I recall one pharmaceutical, at a place I worked, required the six drawn samples to have some wet chemistry performed, either along with or separate from, a UV or HPLC quantification.
In fact, the more modern dissolution apparatus can do a constant, real-time, UV sampling of the dissolution medium -- but I never got to use cool systems such as those. Auto-samplers for the preparation of vials for later HPLC runs directly from the dissolution media are also available. I think, but I could be very wrong, that interfacing an auto-sampling dissolution rig directly to an HPLC auto-sampler is still an experimental dream, at least for mainstream, established, pharmaceutical manufacturers.