Although never widely done, and usually restricted to subordinate members of the hierarchy (e.g., bishops, cardinals) or Catholic sovereigns, such a gesture was intended to be an acknowledgment of the pope's authority and his status as the Vicar of Christ. In the case of a Catholic king, kissing the pope's feet would indicate that the spiritual authority of Christ, as represented by the pope, is infinitely higher than any earthly power wielded by a king. As for the particular gesture of honoring the feet of the man who represents Christ on earth, the gesture does have biblical precedent (cf., John 12:3). These days though, the current Pope likely would gently discourage anyone from trying to kiss his feet.
As a side note, there's a humorous story of an early 20th-century pope who, upon election to the papacy, was approached by the cardinals to be given a ritual sign of their acknowledgment of his authority. By this time the custom was to kiss the pope's ring. One cardinal, overcome by the moment, fell to his knees and bent to kiss the new pope's feet. The pope is said to have told him firmly, "Don't make me give your nose a boot."