Anisette-based drinks rely on aromatic compounds called terpenes for their flavor.
These terpenes are soluble alcohol, but not in water.
The 40 per cent or so alcohol in the drink is enough to keep the terpenes dissolved, but when the drink is mixed with water they are forced out of solution to give a milky-looking suspension.
Absinthe, a similar drink which is based on wormwood and is now banned in some countries because of its toxicity, gives a more impressive green suspension.
Terpenes are responsible for a lot of the harsher plant scents and flavors including lemon grass and thyme.