These interpretations are derived using Gematria- the letter values of the Hebrew alphabet. In Gematric systems, words made up of similar letters or having similar numerical values are believed to be related in some way...these relationships form the basis of the Kabbalistic system.
The Hebrew alphabet differs from Western alphabets in several ways. Each letter of the Hebrew alphabet has a unique numeration as well as a sound, and each letter is also a word with an underlying concept. The alphabet is further divided into three groups according to the method of pronunciation: mothers, singles, and doubles. The three mothers, Alef, Mem, and Shin, represent not only sounds, but elements. Alef, whose pronunciation begins with the expulsion of air in the throat, represents the element of air; words containing Alef also contain the meanings associated with the letter. Mem, the mother letter representing water, is found in many words relating to water, to the womb, and to motherhood. A brief description of each letter can be found here.
Here is a brief example of a Hebrew letter concept used in biblical interpretation: The letter Beth is the second letter of the alphabet. It has the meaning of container or conduit, and is literally translated 'house.' It is considered of great mystical importance that beth is used as the first letter of Genesis. The phrase "In the beginning" in Hebrew is a single word- berashith. Beth is numbered two, and is the first duality- the creation of something distinct. Beth is creation but also separation.
These divisions of letters form the basis for the kabbalistic diagram known as the "tree of life," which is the Kabbalistic diagram of the container of creation.
The tree of life is derived from two twelfth century works, the Sefer Yetzirah, (or book of formation), and the Zohar (the book of splendor). It is a diagram containing ten sefiroth (spheres), or emanations, numbered one through ten, and twenty two paths connecting them. Each sefira is represented by a number between one and ten, and contains all of the possibilities that number represents. The paths between them represent the relationships between the sefiroth. That which is beyond the tree is unity, called absolute limitless light- both ineffable and unknowable by ordinary human understanding.
This tree is further divided into four worlds. These worlds are not places, but levels of consciousness attainable by the student of kabbalah. The four are Atziluth, Briah, Yetzirah, the world of formation, and Assiyah, the world of action.
The divisions on the tree also represent chakras, body parts, astrological signs, stages of life, the human nervous system, and much more- in fact, one can fit all of creation onto the tree. While all of this may seem uneccessarily complex, exploring the relationships between the paths can provide nearly endless insight. Tarot aids in this study by providing a pictoral map for each path.
A standard Tarot deck is divided into four suits of fourteen cards each, plus twenty two 'trumps.' Each of the four suits represents a different element, and each of the numered cards represents that element manifested through number.
There is another reason that the number four recurs in Tarot. Each of the four worlds described above correspond to the Four lettered name YHVH (first found in the chapter of Genesis describing the creation of Adam)