In the Outside, all things exist in an ocean of unreality, made up of substance we cannot comprehend, with its own insane biologies and systems. It is an unborn reality that hungers for the truth, life, and existence of our own creation.
Our reality is protected by the Astral, the plane of human thought, belief, society and civilization. Humanity, by rationalizing the universe, by ordering, quantifying and naming creation--be it through myth, folklore or science--makes our reality more 'real' and reinforces the boundaries between what "is" and what "cannot be". At the very edge of the Astral are the five Pillars of Creation, which incarnate the rules for all living things in our reality: the Fountain defines life, the Keystone defines creation, the Gates define death, the Pit defines destruction, and the World Soul defines interaction and actualization.
These rules limit the Outsiders--they are made limited and less by our rules. And so those Outside try to come in and make our reality more like theirs, the mad chaos where everything and nothing are simultaneously possible and true. Where life and death are not binaries, but a spectrum where you can be both, or all, or none at the same time. Where the orderly progression of time is completely alien and multiple potentialities exist simultaneously. Remember the quote from the Necronomicon: "That which is dead may eternal lie, and in strange aeons even death may die?" This perfectly encapsulates the relathionship between the outsiders and our own thin veneer of reality.
When one thinks of these alien entities, usually it's Dead Cthulhu who springs first to mind. He really does exist, and he lies dead dreaming in the corpse city R'lyeh, where he was bound aeons ago by forces beyond reckoning... but when the stars are right, he will rise again, and all Earth shall tremble at his cosmic might. Because he is bound to Earth, there are countless cults dedicated to Cthulhu, but he's not the only one. Other Outer Gods and Great Old Ones continue to hold Earth in their baleful gaze, including Hastur, Yog-Sothoth, Shub-Niggurath, Dagon, and Tsathoggua. Mad cultists and vile warlocks call upon these twisted entities, and are rewarded with wicked and maddening powers to help them unmake the laws of our reality.
And worst of all of thes is Nyarlathotep, the Black Pharaoh. He is the only Outer God to have slipped past the borders of Creation and incarnated fully in the mortal world. Even in this state, limited by the Pillars of Creation, he is an immensely powerful entity. He is the reason why, despite the efforts of most other factions, cults to the Outer Gods have continued to rise up. again and again. And what makes him even more terrifying than his reality-bending siblings? Nyarlathotep understands humanity. His rot had spread deep into our societies and cultures because he knew our weaknesses, he knew the individuals to target, the knowledge to spread, and the temptations to offer. He wanders the Earth, whispering horrible truths, and we listen.
But these great entities are far from the only intruders from the Outside... indeed, most of their work is done by their servants: the Star-Spawn of Cthulhu, the Dark Young of Shub-Niggurath, shoggoth, deep ones, the Men of Leng, and other, even stranger entities. See, the creatures from Outside don't have to conform to our ideas of what "alive" and "living" are--see the quote from the Necronomicon above. Most famous of these is of course the colour out of space, but it's far from the only kind of strange Outsider that lurks in the strange places of the Outside. There's also a vile martial art which corrupts its practitioners, a spell that slowly breaks down the caster's soul and infects other mages who see it cast, a mental illness that causes one to have visions of a black sun consuming all knowledge and history until they simply cease to exist, a parallel world composed entirely of an empty city with winding alleys that fold back in on itself, an instruction book filled with impossible directions to construct eldritch machines, an old vinyl record with impossible harmonics that will cause one to no longer be able to empathize with other mortals, and other maddening things that Never Were and Never Should Be.
The outsiders want in to our little bubble of reality so very much. They whisper promises of power to those who would open up the gates to the Outside and let their impossible forms through into creation.
And most aren't lying. They're all to happy to reward loyal followers with a fragment of their powers. However, the human body--and reality itself--isn't meant to hold such power, and so it twists them in all sorts of horrible, mind-bending ways. Cultists of the outer-gods have misshapen limbs, bulging eyes, bestial features, and more than a few tentacles spouting from their once-human forms. And that's just the average rank-and-file cultist. The high-priests can scarcely be identified as human anymore, some becoming churches made of flesh and bone, others becoming squamous masses of tumours with a thousand mouths crying out prayers to gods whose names rend the fabric of reality.
And the physical twisting is nothing compared to the spiritual warping that occurs. Most cultists are completely hollowed out, their souls transformed into shards of the Outside that can exist in the outside. Metaphysically, they have become cosmic wounds, their very presence wrong and damaging to our reality.
And what's worse? The same fate can befall those who are sacrificed to the Outer Gods, even if they're unwilling. Once your soul is fed into their horrible, gaping maws, there is nothing left to stop the Outside from reaching in and filling you, taking you over, shattering your mind and your free will. You become nothing more than a shell, puppeted around by the thousand flailing limbs of those who dwell beyond.
That's why Outer God cults are treated as existential threats by the Veil Treaty, and utterly crushed when they are discovered. These groups represent an existential threat to the Earth and the billions of souls who call our world home. Even though we've been fortunate over the course of human existence, it just takes one slip-up for our world to end under a stratosphere of tentacles.