Why does regular exercise increase brain volume?
Well, Erickson et al (2011) attribute the increase in brain volume in the aerobic exercise group to brain-derived neurotrophic factor (BDNF).

Specifically (p. 3020):


  In fact, we found here that changes in serum BDNF levels were
  associated with changes in anterior hippocampal volume; an important
  link because the hippocampus is rich in BDNF, and BDNF levels increase
  with exercise treatments in both rodents and humans. BDNF is a
  putative mediator of neurogenesis and contributes to dendritic
  expansion and is also critical in memory formation. Our results suggest that 
  cell proliferation or increased dendritic branching might explain increased 
  hippocampal volume and improvements in memory after exercise


with the caveat:


  however, increased vascularization (15, 16, 33) and dendritic
  complexity (34) may also be contributing to increased volume