It doesn't, nor does the Church claim the cardinalate to be a biblical, sacramental office. The title of cardinal designates a highly ranked papal assistant and adviser directly appointed by the Pope in a consistory. According to Canon Law, "The cardinals of the Holy Roman Church constitute the senate of the Roman Pontiff and aid him as his chief counselors and collaborators in the government of the Church" (CIC 230).
As a body the cardinals are referred to as the Sacred College of Cardinals, and it is to them (representing the people and clergy of Rome) that the duty falls to elected a new pope (a new Bishop of Rome) upon the death or resignation of the old one.
Although cardinals are often called "princes of the Church," their office is not intrinsically connected with the sacrament of holy orders. In fact, at different times in Church history--most notably during the Renaissance--laymen served as cardinals, although the last one to do so died in 1876.
For more than a century all cardinals had been chosen from among the bishops, but in recent years that has changed a bit. Today there is one cardinal, Fr. Henri de Lubac, who is not a bishop. Pope John Paul II named another priest, Hans Urs von Balthasar, a cardinal, but von Balthasar died a few days before receiving the red hat.
The word cardinal comes from the Latin word cardo, which means hinge and refers to the important role such officials play in the government of the Church.