Reddit user thomascgalvin has an amazing summary of why encountering one of Lovecraft's Old Ones would almost instantly drive you mad.
When you were a child, you were probably taught that you have five senses: sight, sound, touch, taste, and smell. This is true, but it's incomplete.
You have plenty of other senses. A sense of balance. A sense of time passing. A sense of changing temperatures. A sense of where your body is in space. A sense of acceleration. A sense of pain. A sense of loss. A sense of joy. A sense of agency. A sense of belonging.
You perceive the world around you in a dozen different, subtle ways, many of which you can't even name, and most of which you are completely unaware of, unless something goes wrong.
The right (or wrong) stimulus can completely change our psychology. When someone sees the Grand Canyon for the first time, they often report a feeling of overwhelming awe at the sheer size and spectacle. When astronauts return from space, return from seeing this great world of ours reduced to a tiny blue marble out the window, the report a sense of overwhelming concern for, and oneness with, humanity. When someone takes psilocybin or DMT, they report an expansion of their mind that fundamentally alters who they are as a person.
But what about people who have seen the horrors of war? People who have suffered at the hands of an abusive lover? Or people who have suffered something as simple as a car crash, and now break out in a cold sweat every time they drive? The wrong stimulus, applied to our normal, human senses, can irreparable harm us.
When someone sees a Great Old One, the sheer monstrous nature of it, the impossibility, the implacability, the inhuman and inhumane otherness of the thing that they have laid eyes on, is enough to shatter, completely and forever, their sense of who and what and why they are. It's like looking at the splendor of the Grand Canyon and realizing that it will absolutely destroy you, not because it hates you, not because it needs to, but because that is simply what it is.
Just as you were unaware of the fact that you aren't falling until I just mentioned it, you are not aware of the fact that you believe that you are important, that you matter, that you have any hope at all of making a difference in this vast and careless universe until something forces you to confront that fact. And that is what the Great Old Ones do. When you look upon them, you are looking at a power so vast and ancient and amoral that any sense of normalcy you once had will be forever torn away.
And that's just what we can explain with out current understanding of the human mind. Who's to say that we don't have some other sense, a sense of the cosmic, a sense of the psychic, that is triggered by the presence of a Great Old One? Who's to say that some sense, long dormant in the safety of our comfortable little world, isn't woken up as soon as one of those eldritch abominations appears before us?