WiseGeek, the source of Benyamin Hamidekhoo's answer, rightly notes that both the pot and the kettle "turn black with use." That is, they start out a silvery or grayish or coppery color and gradually turn black through exposure to the heat and smoke of the fires or heating elements that they are set above. However, I disagree with WiseGeek's contention that the essential meaning of the saying would be preserved if it were reworded (for the modern sanitary kitchen) as "the pot calling the kettle silver." In my view, one crucial aspect of the original saying is that, in the world of anthropomorphic pots and kettles, being black is an undesirable trait—and that calling a receptacle black, even if the receptacle's blackness is incontestably true, is an insult. Without the element of insult, we might as well have a saying about "the apple calling the strawberry red"—which we don't.
The metaphorical idea at play here is that a clean pot or kettle is like an uncorrupted person, but that through exposure to "blackening" elements—or even perhaps to day-to-day life—the receptacle's original color, like the person's innocence, is lost. It is deeply hypocritical for one receptacle (or person) to criticize another for possessing an undesirable characteristic that (in truth) both of them possess. And as with the blackness of the pot and the kettle, the corruption of both the accuser and the accused doesn't make the accusation of corruption any less of an insult. Having said that, I do share WiseGeek's view that the saying has no intentional racial overtones.
The saying is rather pessimistic, it seems to me. After all, what is more natural than that a pot and a kettle—without the intervention of some vastly more powerful being in the scullery—should turn black in the course of their duties? James Branch Cabell points out something similar with his apothegm: "A man's hands are by ordinary soiled in climbing."
Update (June 27, 2022): A look at the early days of enmity between pots and kettles
As noted by user31341 in a separate answer, a version of this proverb appears in a 1620 translation of Don Quixote:
No more Prouerbs, Sancho, (said Don Quixote) since each of these is enough to make vs know thy meaning, and I haue often aduised thee, not to be so prodigall of thy Prouerbs, but more sparing: but 'tis in vaine to bid thee; for the more thou art bid, the more thou wilt doe it. Mee thinkes, Sir, said Sancho, you are like what is said, that the Frying-pan said to the Kettle, Auant, blacke-browes; you reprehend me for speaking of Prouerbs, and you thred vp yours by two and two.
Another early instance of the proverb occurs in Robert Hayman, "The Third Book of Quodlibets," in Quodlibets Lately Come Ouer from New Britaniola, Old Newfound-land Epigrams and Other Small Parcels, Both Morall and Diuine (1628):
67. To fault-finding more faulty Zoilus. / When others faults thou dost with spite reueale, / The Kettle twits the pot with his burnt taile.
A 1657 translation of Don Quixote (reprinted in 1687) mentions the proverb twice:
Why, what four Proverbs would you have better then these? First, An Humble-Bee in a Cowturd thinks himself a King— and agen, He that Thatches his House with a Turd shall ha' more Teachers then Reachers— And agen, The Horse thinks one thing, and he that rides him another— And agen, Tickle my Throat with a Feather and make a Fool of my Stomach. What a dekins ayls yee, would yee have better Bread then is made of Wheat? They that so easily see a Mote in another Mans Eye, should do well to take out the Beam i' their own, lest the Pot call the Kettle Black-arse.
By my Faith, Sir, quo Sancho, you put me in mind of another Proverb, as pat as a Pudding to a Fryers Mouth—The Porridge-Pot calls the Kettle Black-arse— you reprove me for talking Proverbs, and bring 'em out your self by dozens at a time—But you don't consider, Sancho, quo Don Quixote, that those I speak are to the Purpose—but thou fetchest 'em in by Head and Shoulders without Rhime or Reason.
From James Howell, Paroimiographia Proverbs, or, Old Sayed Sawes & Adages in English (or the Saxon Toung), Italian, French, and Spanish (1659):
The Frying-pan told the Kettle, get thee hence thou black ars.
From Samuel Fisher, Rusticus ad Academicos in Exercitationibus Expostulatoriis, Apologeticis Quatuor: The Rustick's Alarm to the Rabbies (1660):
A Coming together there is for Customes sake in their Best Cloathes, as finely as they can afford to do, when their (supposed) Sabbath comes about in its turn; a sound comes from a Money-Merchants mouth, and enters in at one Ear of the People, as fast as it can get out at the Other; and while it stayes, it Swims in the Head, but sinks not down to Renew the Heart; and some Psalms may be Sung to the Praise of him, whom the Dead in Sin, that live Sin, cannot praise; and so there's an end of the businesse for that day, till it come again; till when, Hell breaks loose, and the Devil is served for the most part all the Week after; insomuch that it is but for the Kettle to upbraid the Pot with its black ugly Hue, for the Priest and his People to make Narratives of the worst of that Naughtinesse, that is found among the very worst of those that are own'd as one in Fellowship by the Quakers, and their Ministry.
From Joseph Hall, "The Woman Taken in Adultery," in The Contemplations upon the History of the New Testament (1661/1679):
It is but just there should be a requisition of innocence in them that prosecute the vices of others. The offender is worthy of stoning, but who shall cast them? How ill would they become hands as guilty as her own? What doe they but smite themselves, who punish their own offences in other men? Nothing is more unjust or absurd, then for the beam to censure the moat, the oven to upbraid the kiln. It is a false and vagrant zeal that begins not first at home.
Hall died in 1656, so the cited passage above is from somewhat earlier than the publication dates would suggest.
From a 1662 translation of Giovanni Torriano, The Second Alphabet Consisting of Proverbial Phrases Interpreted and Illustrated Where Most Necessary : With Pleasant and Usefull Annotations, Italian and English (1662):
Padella, a fryingpan. Dir come disse la pa∣della al paiuolo, i.e. ogni un faccia i fatti suoi, anche fatti in là cul negro, che tù mi tingi, to say as the fryingpan said to the pot, viz. every one about his business, also stand off black arse, thou dost smuch me; the Devil corrects Sin.
The word "smuch" may be an OCR misreading of "smirch," although I don't have access to the original to the original text to check this conjecture.
From John Ray, A Collection of English Proverbs Digested into a Convenient Method for the Speedy Finding Any One upon Occasion (1670):
The kettle calls the pot black a—
The kiln call's the oven burnt-house.
From William Penn, Some Fruits of Solitude in Reflections and Maxims Relating to the Conduct of Human Life, second edition (1693):
386. For a Covetous Man to inveigh against Prodigality, an Atheist against Idolatry, a Tyrant against Rebellion, or a Lyer against Swearing, and a Drunkard against Gaming, is for the Pot to call the Kettle black.
387. Such reproof would have but little Success; because it would carry but little Authority with it.
And from B.E., A New Dictionary of the Canting Crew in Its Several Tribes of Gypsies, Beggers, Thieves, cheats &c. (1699):
The Pot calls the Kettle black A---, when one accuses another of what he is as Deep in himself.
Further thoughts on pots and kettles
Unhappy interactions between pots and kettles go back quite far in English literature. The earliest instance that a Early English Books Online search finds is from a 1539 edition of the Bible (specifically, Ecclesiasticus:
Who so toucheth pytche, shall be fyled withall: and he that is famylier with the proude, shall clothe hym selfe with pryde. He taketh a burthen vpon hym, that accompanyeth a more honourable man then hym selfe. Therfore kepe no familiaritie wt one yt is rycher then thy selfe. How agre the kettle and the pot togither? for yf the one be smitten agaynst the other, it shall be broken. The ryche deleth vnryghtously, and threteneth withall: but the poore being oppressed and wrongously delt withal, suffreth scarce¦nesse, and gyueth fayre wordes.
Evidently, the sense of this passage is that yhe kettle is made of metal and the pot of clay. That at any rate is the implication of the wording used in Andrew Kingesmill, A Viewe of Mans Estate Wherein the Great Mercie of God in Mans free Iustification by Christ, Is Very Comfortably Declared (1574):
There is a good péece of counsel in the .13. of Ecclesiasticus, concerning those with whome we may not win familiaritie, for he sayth: Burthen not thy selfe aboue thy power whilest thou liuest, and company not with one that is mightier and richer than thy selfe: for howe agrée the Kettle and the earthen pot togither? For if the one be smitten against the other, it shall be broken. Wherefore in thys behalf an equalitie were a goodly propertie.
I was also interested to learn that pots were sometimes set inside kettles of water in order to separate them from direct contact with fire, perhaps something like a double boiler. For example, from a 1565 translation of FGirolamo Ruscelli, The Secretes of the Reuerende Maister Alexis of Piemount Containyng Excellent Remedies Against Diuers Diseases, Woundes, and Other Accidents:
Take oyle of the eldest you can finde, and boyle it the space of an houre, & for euery poūd of the said oyle, put in .l. scorpiōs, or as many as you can get, put all this in a pot vncouered, the which pot you shall set in a kettle or caudron of boylinge water, vntill the thirde part of the oyle or somewhat lesse bee consumed.
Of possible relevance here to the proverb of "the pot calling the kettle black" is the fact that, in the scenario described above, the bottom of the kettle or cauldron would be blackened, but the bottom of the pot would not. In effect, the kettle shields the pot from getting blackened. If this were the original scenario envisaged by the proverb, the pot's criticism of the kettle's appearance would be doubly inappropriate because it would be turning the kettle's interceding between the pot and the fire (for the pot's benefit) into a source of mockery.
However, this speculative interpretation of the proverb is not supported by the history of the expression, in several ways. First, the earliest form of the proverb (from 1620) involves a frying pan and a kettle, not a pot and a kettle. Second, when the pot dos make its first appearance (in 1628), the instance has the kettle twitting the pot for its "black taile." And third, there is no reason to suppose that the pots in any occurrence of the proverb re imagined to be earthenware, rather than metal and thus kept from direct (and blackening) contact with fire and smoke.
The proverb "the pot calling the kettle black" has early counterparts in Spanish and Italian. It may very well have originated in one of those countries, albeit with a frying pan in place of a pot.