In the Bible, antichrists are simply false preachers lead by Satan. So where does the above belief come from? It's prevalent in pop-culture depictions of Satanic cults, such as the film Rosemary's Baby.
At first it may truly seem that it is not in accordance with the Catholic Faith to maintain that children can be begotten by devils, that is to say, by Incubi and Succubi: for God Himself, before sin came into the world, instituted human procreation, since He created woman from the rib of man to be a helpmeet unto man: And to them He said: Increase, and multiply, Genesis ii, 24. Likewise after sin had come into the world, it was said to Noe: Increase, and multiply, Genesis ix, 1. In the time of the new law also, Christ confirmed this union: Have ye not read, that he who made man from the beginning, Made them male and female? S. Matthew xix, 4. Therefore, men cannot be begotten in any other way than this. But it may be argued that devils take their part in this generation not as the essential cause, but as a secondary and artificial cause, since they busy themselves by interfering with the process of normal copulation and conception, by obtaining human semen, and themselves transferring it.
Demigods and children of the devil were a thing in the literature of antiquity and then again in the Middle Ages, being one of the major issues dealt with in the Malleus Malleficarum (as mentioned above). The idea pushed by the author of that text was that demons could not be impregnated or impregnate mortals directly, but instead would act as go-betweens: a succubus would have sex with a man, receive his seed, and then change into an incubus and deposit the seed into a woman - and in this way perpetuate the families of witches.
The older idea of the physical union of devil (or an equivalent entity) and human was revived in the late 19th century Victorian fiction. Arthur Machen's "The Great God Pan" published in 1895 had a virgin give birth to a kind of succubus after receiving a brain operation where she "saw the Great God Pan"; more spectrally, Aleister Crowley wrote the Moonchild (1917) about a woman to be impregnated by magical, incarnating a spirit. Both stories deal with a kind of deliberate invocation and subversion of the annunciation of Mary and the conception of Christ. As Machen puts it:
ET DIABOLUS INCARNATE EST. ET HOMO FACTUS EST.
(It is perhaps worth mentioning that both of these writers were associated with the Hermetic Order of the Golden Dawn, which had some sex magick at the upper end of its curricula, and some other writers in the same occult circle explored similar concepts to a lesser degree.) Both of these stories were inspirational to other writers; H. P. Lovecraft cites Machen in "The Dunwich Horror" (1929), and Clark Ashton Smith was inspired by Machen to write "The Nameless Offspring" (1932).
Neither of these - or most of their successors - deal with the eldritch offspring as "antichrists" explicitly, although the parallels and parody of the supernatural conception are there, and all of the offspring are to certain degrees inhuman and possessed of supernatural powers, harkening back to the older concept of demigods et al.
Rosemary's Baby (1967) by Ira Levin made the Satan-impregnates-a-woman connection explicit, and popular; but it is noteworthy that he never uses the term "antichrist" in either the book or the immensely popular film that followed in 1968. What Levin does do is make explicit how this book is an inversion of the Christian text:
He came up from hell and begat a son of mortal woman.
Satan is his father, and his name is Adrian.
He shall overthrow the mighty and lay waste their temples.
He shall wreak vengeance in the name of the burned and the tortured!
He chose you. Out of all the women in the whole world, he chose you.
He wanted you to be the mother of his only son.
The success of Rosemary's Baby inspired copycat films and works, including, ironically enough, a film adaptation of The Dunwich Horror (1970) and finally The Omen (1976) which finally made the antichrist connection explicit. Outside of film, you have media like Marvel comics Son of Satan (1973), which sort of combines the basic concepts (Satan physically impregnates a human woman, the progeny has supernatural powers), but the whole antichrist angle doesn't explicitly come into play until decades later - Marvel followed Jack Kirby's The Demon (1972) in making their infernally-powered character a hero, instead of the villain of the Omen films and much later media.