Does the growth in size of hippocampus of London taxi drivers affect their net number of brain cells?
The study you linked was very interesting, and there is a reason that the authors of this paper and many others like it refer to grey matter density or grey matter volume.

From another study on grey matter (density), Neurolinguistics: structural plasticity in the bilingual brain,


  Whether grey-matter reorganization in this region is related to changes in neuropil, neuronal size, dendritic or axonal arborization will be revealed by methods other than whole-brain magnetic resonance imaging.


Meaning, there's really no way to know without autopsy and histological studies whether this means more neurons, increase in neuron size, more dendrites, more connections between neurons, etc. So, the answer to your question is, no one knows exactly.

Histopathological studies are usually done on diseased patients (e.g. Alzheimer's), not healthy taxi drivers.

However, one mouse study of loss of hippocampal grey matter did correllate MRI studies and histopathological studies, with the result that


  no changes in the number or volumes of the somas of neurons, astrocytes or oligodendrocytes were detected. A loss of synaptic spine density of up to 60 % occurred on different-order dendrites in the ACC and hippocampus...


Loss is not the same as gain, however, so the definitive cause cannot be ascertained yet.

Stress-Induced Grey Matter Loss Determined by MRI Is Primarily Due to Loss of Dendrites and Their Synapses