Nihilism is simply belief in nothing, no more, no less; it isn't a Brown Note which makes a person Go Mad from the Revelation, pushes him onto the Despair Event Horizon, turns his life into an instant Cosmic Horror Story, converts him into an omnicidalCard-Carrying Villain, or automatically turns him wangsty or depressed. There are also many interpretations for nihilism. For example, Metaphysical nihilism is a literal belief in nothing. Moral nihilism is the view that morality does not exist as something inherent to objective reality; therefore no action is necessarily preferable to any other. For example, a moral nihilist would say that killing someone, for whatever reason, is not inherently right or wrong. Existential nihilism is saying life is meaningless and insignificant — the form of nihilism which is often portrayed in this trope.
However, this is a true Mind Screw. How does one believe in nothing, and yet believe this belief is truth? Because truth is something, but...
Generally they don't. Stuff about the illusion of consciousness and the inability to perceive the universe objectively is a big part of a lot of nihilism's various varieties; they're just going with an idea that may or may not be true but is the most likely outcome based on their scientific/metaphysical/whatever beliefs. This is also why it's not a magical brown note that makes people immediately commit suicide; pragmatism is still a thing, and it's still completely feasible to live a normal life while maintaining that the universe has no objective order to it. Recognizing your reproductive instincts as a biological mechanism doesn't stop you from feeling them, for example.