LSD, like other psychedelic drugs (Psilocybin, MDMA, DMT, ...), floods the brain with (mimics of) neurotransmitters, enabling it to make (almost arbitrary) connections. Hence the synesthesia. It's "therapeutic" effect stems from the fact that the brain/mind/consciousness is leaving the treaded paths it usually takes "automatically" and forms (forces) new trains of thoughts.
Hence the fascination: People are used to their brains working in a particular way, sorting your experiences into "proper" categories, and are stunned when they realize that they can actually process information in completely different ways as well. This is what people mean talking about "filters" being "removed". Aldous Huxley popularized this idea in "The doors of Perception".
Different medications like SSRIs have this effect as well, but by magnitudes less strong.
It's a torrent of thought(fragments) that your brain switches through on this drug. There might be some in there that actually help you cope with a problem you had in your life, but there are lot of "useless" bits as well.
So what taking LSD does is giving you a perspective you haven't had before. This might be of therapeutic value, but so can be other experiences you haven't had before. Like living in a monastery in Tibet. Getting a baby. Seeing a fellow soldier getting killed in the field. Not sleeping for 70 hours.
I'm sceptical of the praises because:
* LSD doesn't make your prior brain-structure go away. It softens it and forms new paths, but chances are high that you go back feeling the same and thinking the same as before. True therapeutic progress is always slow and iterative, because that way it is stable and lasting. Slamming the psychedelic hammer onto your mind knocks you out of your path, but the experience can't be integrated that well because it usually is too random. Also the iterative approach (meditating, behavioural therapy) makes your brain actually start to produce the neurotransmitters needed to form the desired thoughts.
* People usually feel really well for some time after taking this. This is logical, because they realized that their mind isn't as immutable and frozen as they were afraid it is. Also on strong serotonergic agents like LSD you also experience bodily effects like low to moderate fever (which you don't feel cause you're somewhere else). This can culminate in a serotonin-syndrome [1] [2]. When coming down from this condition it's naturally that you feel well, like you would "coming down" from food-poisoning.
What I find really interesting are two common emotions/feelings that people on psychedelic drugs experience:
* Spontaneous insight: that all the things they are experiencing are "true", "right", "eternal". Also "sacred" or "holy".
* All the things around are alive, vibrant, conscious.
I would like to know what in the mind actually produces the feeling of "truth" and what in the mind discerns between "conscious" and "unconscious" things.
[1] https://en.wikipedia.org/wiki/Serotonin_syndrome
[2] This is actually the main reason for people dying while partying on MDMA/ecstasy: their organs overheat.