Short version: Diacetyl is a byproduct of valine biosynthesis in yeast.
Long version: Yeast need amino acids for growth and protein synthesis. If the yeast can't assimilate sufficient amino acids from the wort, it must synthesize its own. Valine (,leucine and isoleucine) is/are synthesized through the ILV-pathway. In the case of valine, pyruvate (a key intermediate in several metabolic pathways) is first converted to alpha-acetolactate (AAL) through the action of the acetohydroxyacid synthase (AHAS) enzyme. AAL is then further converted to valine through a series of reactions. Typically, AAL is produced in excess, and it leaks out of the cell into the wort (the actual reason for this is still unknown). In the wort, AAL is then converted into diacetyl. Valine feedback-inhibits the acetohydroxyacid synthase (AHAS) enzyme, that converts pyruvate into alpha-acetolactate, which means that the more valine that enters the cell, the less AAL is produced and thus the less diacetyl is produced as well. Be careful though with raising the FAN level too much (i.e. adding too much amino acids), as it can actually have the opposite effect of increasing diacetyl production (as well as higher alcohol and ester production)! Why is that? Well, one would think that a high FAN value would automatically result in less diacetyl, since more valine is present in the wort, but this is not always the case. For the valine to inhibit the AHAS enzyme, it must be transported into the cell. Amino acids are transported into yeast cells through various transporters in the membrane. Valine is taken up through several transporters, but none of them are specific for just valine (i.e. they transport other amino acids as well). When the amino acid concentration of the wort is increased, the competition for these transporters increases as well, and as a result less valine is actually taken up (adding pure valine or an amino acid mix high in valine would of course not result in this). A good target FAN value for optimal fermentation yet low diacetyl formation is in the 150-300 ppm range.
For more information, see this recent review:
http://onlinelibrary.wiley.com/doi/10.1002/jib.84/abstract