Belief is gained when supernatural, fantastic, or super-science abilities, creatures, activities or weapons/equipment are seen or encountered in public. The bulk of the Monster Control Board is composed of personnel who are trained just to deal with, and prevent the spread of, belief in the supernatural. In addition, the ancients built numerous structures with the purpose of bleeding off the belief energy, without it causing weaknesses in the fabric of reality.
Belief in the supernatural causes the fabric of reality to weaken, empowering magic and monsters in general. More monsters will appear, with greater frequency, and generally with greater power. Magic, and magical abilities, will gradually gain power and potency. More creatures will find themselves able to wield magical abilities. Eventually 'things' locked away outside our dimension will be able to force their way in, causing untold horrors. Dogs and cats, living together. Mass hysteria.
The MCB has no real way to drain off belief without just straight up murdering the people that have encountered the supernatural (which they are legally allowed to, if they see the need). What they do instead, is use subterfuge and misdirection to convince people that the impossible things they saw was something mundane (that green explosion was a methane explosion, or that giant purple squid was a prop for a movie, or that building the had armed agents around it that blew up was an anarchist/communist/criminal hideout). Failing that, the MCB agents will make it painfully clear to witnesses that they cannot EVER tell ANYBODY about what they saw, under threat of various penalties (which range from prison time to being committed to an insane asylum to literally being murdered).
Belief energy drains away slowly naturally, and this is enhanced by a variety of artifacts/structures the ancients have built. One of the MCBs main goals is to uncover how to drain this energy faster.