Do I need some special psychic abilities to read the Tarot?

No you do not.  Everybody has some latent psychic ability within themselves.  The point is to develop this ability while focusing on your specific personality, the appeal of your cards, and the person you are reading for.

It is of course best if you begin by doing readings only for yourself.  This will help you to get to know your cards and spreads thoroughly.  As you become increasingly good at interpreting your cards, you'll become more confident, and you can begin doing readings for others.

You should give yourself time and enjoy the process of getting to know the cards.  Reading them is a combination of knowing each card meaning and your intuition.  Ultimately, the answer is that anyone who wants to can read the Tarot.