Why the Greeks Believed the Muses Existed
From Hesiod's Theogony, detailing why the Muses were born:
"They were born on Pieria after our Father Cronion
Mingled with Memory, who rules Eleutherai's hills.
She bore them to be a forgetting of troubles,
A pause in sorrow."
And so their Mother, Memory (Mnemosyne), had daughters whose purposes were to distract from her own darker qualities. The Greeks believed that the function of the Muses, who were the inspirations/guardians of:
epic poetry (Calliope)
history (Clio)
love poetry (Erato)
mimicry (Erato and Polyhymnia)
music (Euterpe)
tragedy (Melpomene) (though she also sang)
sacred poetry (Polyhymnia)
geometry (Polyhymnia)
meditation (Polyhymnia)
agriculture (Polyhymnia)
dancing (Terpsichore)
comedy (Thaleia)
playful/idyllic poetry (Thaleia)
astronomy (Urania)
Thinking from the purely historical stance, and with the assumption that the Greeks fabricated their religion over time based upon human needs rather than there actually being these Greek gods and goddesses (just an assumption for the sake of pursuing this line of thought, of course!), one observes that they needed these specific things in order to construct meaningful lives, and not always be in a dredge for remembering troubles past. These were the methods with which they made themselves joyous.