I'm writing a story in which witches from the "higher realm" visit the "lower realm" (Earth) with a consistent frequency. However, most of, us mortals do not know it. How can I accomplish such a cover up, and why would it even be necessary to cover up in the first place?
why would it even be neccesary to cover up in the first place?
Early modern witch trials: "An estimated total of 40,000-60,000 people were executed during the witch trials."
Modern witch-hunts: "While an unknown problem in vast parts of the Western population, body-counts of modern witch-hunts by far exceed those of early-modern witch-hunting."
I'm writing a story in which witches from the "higher realm" visit the "lower realm" (earth) with a consistant frequency. However, (most of) us mortals do not know it. How can I acomplish such a cover up
The usual conceit is that: (1) witches can "pass" for human, (2) they refrain from using their powers when mundane people who can't be trusted are able to observe them as a universal code of ethics, (3) they self-enforce the code of secrecy if it is violated, and (4) they use whatever measures are necessary, including brain washing by magic, to insure that unreliable muggles are prevented from sharing their experiences with others in a way that will be taken seriously.
This is the approach taken in at least 90%+ of the contemporary fantasy genre (in all fictional media), and also in kindred genres such as those involving aliens (see, e.g., the TV series Roswell and V, and the movie series Men in Black), gods and demigods in a modern setting, and those involving fairies, angels, demons, replicants, androids, superheroes, psychics and mutants.
The first element ("passing" for humans) often takes some additional effort on the part of the non-humans. One decent example of a case where some serious effort is necessary for witches to "pass" as human is The Witches (1983) by Roald Dahl, in which a hidden corps of witches among humans has to engage in some previous serious (basically non-magical) efforts to disguise themselves to go unnoticed. Elaborate non-magical disguises were also commonplace in Men in Black. In Stephanie Meyer's novel The Host, the non-humans hid inside human beings. In the modern reboot of Battlestar Galactica the robots posing as humans were carefully modeled to look like humans. In the case of much of the fiction involving fairies, the fey use what is usually called a "glamour" to conceal their appearance to the average human's mind.
If you can get your hands on an old copy of Stalking The Night Fantastic, a role playing game from the 80's, you will find it immensely helpful. It is basically a users guide for a new recruit to a secret organization, Bureau 13. That organization is charged by the governments of our world with a singular mission, to hide all supernatural denizens and their activities from the mundanes.
From subterfuge techniques to a list of infiltrated sups who live among the mundanes and can be of great help to Bureau's goals, that game manual has everything you need to defend your visitors from the higher realms.
The first question you need to ask yourself is "who is covering it up?"
Either the higher realm wants to remain secret and as such keeps it's members in line or someone on Earth wants it to remain secret.
The higher realm may want to keep it secret because humans are pretty violent and xenophobic. If human found that there were super powered creatures living among them operating in secret, they would hunt them down. If the higher realm felt that they would lose such a fight, they would try to keep their existence a secret.
The Dresden Files by Jim Butcher uses this idea. Vampires, witches, fairies etc all exist but use magic to hide from humans. While humans are weaker than them, there are so many humans with so many weapons they fear humans hunting them down.
The other option is someone on Earth want them to be a secret. If magic actually exists and powerful beings on other planes of existence actually exist then the possibility of "doing a deal with the devil" becomes an actual possibility. The church and/or government would want it to remain a secret so people don't go looking in places they shouldn't. They would employ agents to hunt down threats and silence witnesses by any means possible.
Monster Hunter International by Larry Correia operates like this. Dark forces gain power from human belief so the less people know, the less powerful the forces are. The governments of the world have their own agents to hunt down threats and silence witnesses but they also pay private contractors bounty for every monster they kill.
How can I accomplish such a cover up,
Misdirection. Clearly people are aware witches exist, and there are witch hunters. So expend some efforts making sure what they're looking for isn't what witches actually are.
Publish stories about witches as warty old ladies riding brooms, vulnerable to fire or water. Convince witch hunters to look for certain marks and paraphanalia as sure fire signs. Scam bored housewives into reviving a pagan religion.
and why would it even be necessary to cover up in the first place?
Make the witches feel vulnerable in the lower realm.
They come for a reason, but are still out of their familiar environment, and maybe they can only 'cross over' at a certain time of year*. There are so many mundanes with their strange devices, any of whom could be or become witchhunters, which are at least annoying to deal with. Maybe there is also only so much mana to draw on for protection, and using it up is flashy (or spooky), and that only ever makes things worse. Perhaps there is actually even some form of witchbane. The best defence against fanatical witchhunters is thus not to attract attention.
$^*$ Easter - that other time is pure misdirection.