Biology or Biochemistry

The general answer to how cyanobacteria or other species produce toxins, it's by transcribing the 'recipe' from their DNA/RNA. These neurotoxins are peptides assembled from amino acids, they're synthesized by the usual protein synthesis methods available in the cell afaik.

The general answer to why some species produce toxins, it's thought ecologically that toxins serve as a defense against predation. Animals that eat them, will be sick or die. The code for neurotoxins in cyanobacterial genome is an advantage to their survival.

It's also true that specific environmental conditions cause 'blooms' of cyanobacteria which are toxic. The conditions are good for their growth: and some conditions may favour synthesis of the toxin as well (This is the case in plants ie medicinal plants produce more or less of their specific medicinal constituents in different environmental conditions).

There are some books on the subject you might want to read for a more complete answer, that give an overview of the studies that have been done, for example http://books.google.ca/books?id=bdxdETQozWgC&pg=PA617&lpg=PA617&dq=cyanobacteria+neurotoxin+ecology&source=bl&ots=DGR4D11_Os&sig=ydtSl9_8hqKzPj-BUNCuqnFFyKA&hl=en&ei=xueZSte4K8qwmAf2j6ixBA&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false

(By the way, the name 'cyanobacteria' refers to their colour (blue = cyano). The toxins have nothing to do with cyanide.)