I grew up in a Baptist household (personally Agnostic now) My mom spouted for years about this idea that we were going to all be micro chipped and one of them will be the "Anti-Christ" I'm sure it was probably some crazy person in the 60's-70's.
So, perhaps I'm not the greatest person to answer this. I'd also discourage asking a question as modern as this, on an historical subreddit, in fact you'd probably get a straighter answer from a Christian subreddit, but I'll do my best.
The reasoning behind it, though baffling to those who do not know about revelation/mark of the beast, does have some credence if you're knowledgeable in scripture as a Christian.
The line that promotes the theory that microchips are the mark of the beast, is this.
"Also it causes all, both small and great, both rich and poor, both free and slave, to be marked on the right hand or the forehead, so that no one can buy or sell unless he has the mark, that is, the name of the beast or the number of its name. This calls for wisdom: let the one who has understanding calculate the number of the beast, for it is the number of a man, and his number is 666."
The reason why this is used as proof that the microchips will be the the Mark of the beast, is because it is most likely that the future will require microchips for payment, and they'll most likely be implanted in your right hand and perhaps in/on the forehead (though I don't see why they wouldn't be allowed on someone's left hand).
Another interesting bit of trivia, is that the King James Version of the Bible, says that it will be marked in the right hand, not on it. Which makes it even easier to claim that the microchip will be the mark of the beast.
That is essentially where the logic behind it stems from. There are plenty of other verses that talk about the Mark of the Beast if you want more information. Personally I find it interesting to say the least, since the connection is fairly obvious regardless of your stance on religion.
I hope this answered your question, have a nice day.
I'd love to follow up on /u/NaivePretender's answer with some more historical context, both pertaining to the biblical Mark/Number of the Beast and its resurgence in modern conspiracist culture.
As was pointed out, the Mark of the Beast is a concept out of Revelations, and has since, understandably, become quite a powerful and elastic symbol, as it so easily lends itself to application in different political and social situations. Is the local bishop forcing admitted heretics to wear yellow crosses on their clothes, like in Languedoc after the Albigensian Crusade? He must be the Antichrist! Is the pope instating new rules for the workwear of clergymen? Certainly the Antichrist. It should come as no surprise that history has seen A LOT of Antichrists.
The most likely explanation for this passage in Revelations, though, is that it was meant as a coded polemic against the Roman emperor, either Nero or his successor, Domitian. The Number of the Beast, 666, can be read as a numeric code for the name Nero, and the 'Mark' that one is forced to wear to participate in the economy most likely refers to Roman currency - which would bear the mark of the emperor. This 'code' would probably have been recognizable to anyone reading Revelations at the time, but the haze of time has made it quite a lot more opaque to a modern reader, as is true for many biblical sources.
Anyways, the connection between microchips being implanted in people's skin and the Mark of the Beast was a fairly obvious one, and one that is directly connected with the burgeoning New World Order conspiracies of the late 20th century. In short, these are a large group of loosely-connected conspiracy theories - mostly Christian, mostly millenarianist (that is, apocalyptic), often far-right - positing that a secret cabal of wealthy elites (often, specifically Jewish elites) are trying to gain control of the world through a panoply of organizations and goverment offices (I'm sure you've been hearing a lot about the 'Deep State' in recent years), specifically with the aim of instating a 'New World Order', usually meaning a global superstate with controlled population growth, little personal liberty and heavy surveillance.
Most likely, this specific part of the theory originated with Mary Relfe in the 1981, though her ire was focused on readable bar codes - but the idea quickly proliferated within conspiracist milieu's from there, and was pretty soon associated with microchips as well. According to Michael Barkun, who is also my primary source here, the fundamentalist Christian personality Texe Marrs might have been the first to make that leap sometime after Relfe, though it's hard to say exactly where it originated.
Hope that helps!