Even if you don't believe in magic, tarot cards can "work."
Many traditional forms of divination are straightforward. You throw runes that spell an answer, poke a finger at random into a bible, or feel around in a handful of animal guts. A question is asked, and an answer is given. Someone may or may not stare into a crystal ball or bowl of water.
By comparison, the tarot cards are far more than a simple fortune telling tool. Many people use them to find answers about the future, and those answers can be found. But the tarot cards also act as a set of psychological prompts, which ask the questioner to consider things in a broader scope. Tarot cards don't just "tell your fortune," they also ask you to consider what the future means, what forces will shape it, and what part you yourself will play in bringing that future to bear.
There is no single explanation for how the tarot cards work, meaning the mechanism by which you end up drawing the correct (i.e. relevant) cards from the deck.
The most mundane and boring explanation is that there is no grand order, that your brain finds patterns in the cards where there aren't any, and that (if someone else is reading your cards for you) there is an element of the classic "cold read" happening. But even if this is the case, the tarot cards' symbolism can be helpful, prompting you to consider the larger or deeper meanings behind the events which are happening in your life.
Some people believe that you subconsciously shuffle the deck such that the relevant cards end up on top. Others believe that the deck is like Schrodinger's cat, and that the cards are blank until just before you turn them over. Still others believe that a guiding force - be it angels, spirit guides, or what have you - move the cards around into the proper order as you shuffle.
Telling someone's fortune is an interesting proposition. It presumes that the future is fixed - otherwise, there would be no way to tell it. I can't tell you the plot of a movie that hasn't been written. But I can tell you the plot of a movie that I have seen, but which you have not yet watched.
But if the future is fixed, what happens when you tell someone about it? If I tell you that you will slip on a banana peel, you will look down, see the banana peel, and step over it. If your actions change the future that I just told you about, then how can we say that the future is fixed?