How does activation of adenylcyclase lead to increased cAMP and diarrhea?
The crypt cells of the gut can actively secrete electrolytes (ions) via special channel which finally leads to water secretion. This is usually tightly controlled, since uncontrolled loss of water is obviously not good for the cells.
Have a look at this schematic figure, (from this webpage):




Sodium, Potassium and Chloride ions are co-transported into the cell,
sodium and potassium are transported back out via other channels.
If the adenyl cyclase is activated (for example by bacterial toxins,
cholera does this), then cAMP is made.
cAMP aktivates Protein Kinase A (PKA) which in turn phosphorylates the the Cystic fibrosis transmembrane conductance regulator
(CFTR)(this channel is also important for cystic fibrosis) which leads
to the secretion of chloride ions from the cell. This builds up a
negative potential across the membrane (positively charged ions are
missing).
This leads to the secretion of sodium ions through the tight
junctions, resulting in the netto secretion of NaCl.
The secretion of salt builds up and osmotic gradient across the
membrane, which leads to water secretion and can also lead to hyponatraemia.


To overcome this problems, the so called "oral rehydration therapy" is used. The solution contains of an isotonic solution of sodium chloride and glucose. This is taken up by the sodium-glucose symporter (a special class of transporter, that transports two molecules in the same direction) and builds up a gradient which is directed in the opposite direction of the diarrhoea, helping to fight dehydration.