How exactly does marijuana damage brain cells?
There is no damage of brain cells, as in the context of lesions in neurodegenerative disorders. There are, however, two mechanisms attributed to the memory loss associated with cannabis/marijuana and these are purely physiological responses.


Astrocyte mediated: CB1R (cannabinoid receptor) in astrocyte gets activated and promotes glutamate release. This glutamate induces Long Term Depression in post synaptic neuron. 
Synaptic currents can cause both LTP and LTD. The properties of the synaptic current wave dictates the outcome. Please refer to review articles on LTP and LTD for more information.
GABAergic interneuron mediated: CB1R activation in GABAergic interneurons (inhibitory neurons) causes activation of mTOR pathway and promotes protein synthesis. CB1R activates mTOR through PI3K/Akt.