Quick Answer: Tarot cards were invented in 15th-century Italy as a parlor game called Tarocchi (similar to Bridge). They were not used for divination until the 1700s, when occultists attached mystical meanings to the archetypal artwork.
The Invention of Trumps
Standard playing cards migrated from the Islamic world to Europe in the late 1300s. Around 1430 in Northern Italy, wealthy families added a 5th suit of 22 picture cards (the "Trionfi" or Triumphs). These picture cards acted as permanent trump cards during gameplay. This 22-card suit eventually became what we now call the Major Arcana.
The Shift to Mysticism
For 350 years, Tarot was just a card game. But in the late 18th century, French clergyman Antoine Court de Gébelin published an essay claiming (incorrectly) that Tarot cards contained the lost magical secrets of ancient Egypt. Following this, occultists like Jean-Baptiste Alliette popularized using the cards for fortune-telling (cartomancy).
Tarot as Psychological Mirrors
Today, a growing movement uses Tarot purely as a psychological tool, heavily influenced by Carl Jung's concept of archetypes. Pulling a random card introduces chaos into your thought process. The card's universal archetype (The Fool = new beginnings, Death = sudden change) prompts the brain to reflect on personal situations from an outside perspective.