I'd like to prefix this as _anecdotal_ as it is only some speculation I've done on the matter, but here is my take on it:
The brain is in the end a neural network, and like all such devices, can get stuck in various local minima that are difficult to get out of. An old habit that has been reinforced through many years can take a considerable "shock" to dislodge. All the little cognitive and behavioral biases that one accumulates through their life get in the way in very subtle forms that are entirely sub-conscious.
What psychedelics like LSD, psilocybin and others can do, is remove or downregulate these internal filters for some time. To give an example: the visual cortex normally applies a filtering mechanism to what the eyes see and correct problems. It acts as a common-sense filter to remove improbable or impossible information and augment, perhaps even add some extra detail to save the conscious part from being swamped with ridiculous data. Mostly it just lets the sensory input through with little synthetic stuff. This means that the visual cortex has to have some sort of model of reality that it builds over time (with feedback from the consciousness). This filtering mechanism is one way mostly, the visual cortex modulates the input and adds information here and there, and then reinforces the internal model with feedback form the consciousness.
Now, under psychedelics, one gets all the output that the visual cortex can generate together with the actual real data from the eyes. The previously subtle filter applies the full transformation described by the internal model to the sensory data. This results in wacky outlandish shapes and impossible landscapes as reality is modified fully with all the information that the visual cortex has accumulated. Quite like the bizarre images generated by artificial CNNs when ran backwards!
So to arrive at my point, I posit that psychedelics can (not guaranteed of course), remove the long standing subconscious filters and biases that one has built over their life and offer the raw perspective on things. They remove all the useful illusions that the brain pulls over our eyes to get through our days. A thing that one tends to lose during a "trip" is control over the thought process, some call it ego death, a profound sensation of being able to observe one's thoughts from outside - as if unburdened by one's own insecurities and from an objective view. It isn't unpleasant at all, if anything it is liberating. It is always difficult to face one's own shortcomings, because of the importance that we place on the ego. Once that burden is removed, it becomes an experience of resignation and healing.
I want to avoid ascribing religious or transcendental characteristics to the process, far from it. It's a mere result of some very pleasant time spent being free from the tyranny of the ego and the subsequent introspection.