The short answer is: they don’t.
But they do, when you’re trained to use them, trigger something valid but inexplicable, something that every tarot reader will tell you is real and reliable and useful: Intuition.
Some tarot readers study for years, using books and gurus and practice to hone their access to information that surfaces from the unconscious. Other readers are able to use the cards with less formal training. The cards serve the reader in the same way that holding a physical object such as a watch or other personal belonging can trigger a psychic’s clairvoyant (seeing things) or clairaudient (hearing things) information. The tarot reader is drawn to the cards as a tool for accessing intuition, and over time learns to rely on his or her trained intuition with or without using the cards.
Intuition is an innate human gift. Some people have more access to and reliance upon it than others, the same way some people have more musical or artistic ability than others. But pretty much everyone who isn’t physically disabled can pick up a pencil and draw a recognizable tree; most people can identify a known song even if they can’t sing it in tune. In just that way people have “gut” feelings, or they “just know” about something, and with practice those feelings can be evoked at will and trusted.
I know that whenever my intuition prompts me to do something I’m always sorry if I ignore it. And sometimes it’s positively odd what I get prompted to do: take a couple of large garbage bags to my sister’s house, for instance, only to get there and see that she’s spent the morning clearing out the garage and has run out of garbage bags, and the ones I brought are exactly how many she needs. And no, she hadn’t told me she was planning to tackle the garage.
How do you access your own intuition? Just ask. Seriously, that’s all it takes. Ask yourself, “What should I do?” or “How do I really feel about this?” or “Should I actually go ahead with this idea?” and see what your unconscious brings to the surface for you as an answer. And then listen to the answer and, most important, heed that inner voice. The more you practice asking and listening and heeding, the more your intuition will strengthen.