Okay, sure. Anybody can look at a king these days, if they can find one - but why a cat?
As you may imagine, the phrase means "any lowly person has certain rights before a superior," more or less. At phrases.org.uk, the saying is attributed to a collection of English proverbs dating from 1562. They do note that the saying's origin is unknown, but I have found a possible one -- and it involves Albrecht Durer.
It seems that after Durer completed his massive project "Triumphal Arch of Maximilian" in 1515, the task of engraving the work went to a fellow named Hieronymus Rosch of Nuremberg. (Not Hieronymus Bosch. I checked. Very odd.) The "Arch" had been done under the patronage of Emperor Maximilian I, who liked to drop by Rosch's house to see how things were going. On one of his visits a number of Rosch's pet cats ran into the room where the Emperor was visiting - and it's said that's where the original cat got its look at a king. I got that from page 90 of
Artist Biographies in Five Volumes, vol IV: Dürer. Rembrandt. Van Dyck by Moses Foster Sweetser (Boston: Houghton, Osgood, and Company/ Cambridge: The Riverside Press, 1860).