Edema factor is adenyl cyclase which is activated only inside the target cell. It leads to intracellular accumulation of cyclic AMP resulting in edema. How?
Edema factor is a high efficiency adenylyl cyclase, an enzyme that converts ATP to cyclic AMP (cAMP). cAMP is an important second messenger in many pathways, which you can read about in Albert's Molecular Biology of the Cell. In order to be an effective second messenger, it must be tightly regulated. Several bacterial toxins disrupt this regulation, by creating an excessive amount of cAMP directly (B. anthracis EF), forcing the local activator of adenylyl cyclase to stay on (V. cholerae CT), or preventing the local inactivator of adenylyl cyclase from turning it off (B. pertussis). Some of these pathways are well understood. The pathway between cholera toxin and lethal loss of intestinal fluid, as mediated by elevated cAMP, is particularly well understood. The pathway between edema factor, edema toxin, cAMP, and tissue/organ edema is not well understood, as illustrated by this lovely figure, from this review:
Note the nice question mark between Edema Toxin and Edema
It could be mediated by protein kinase A (PKA), and probably is at least partially. It could also be mediated by EPAC and the small G-proteins Rap-1 and Rap-2 (and probably is, at least partially). It could even be mediated by the loss of ATP, since the anthrax ET is so efficient at catalyzing the conversion of ATP to cAMP.