Psychedelics cause old neural pathways, where memories and events are stored, to become activated in an attempt to make sense of the psychedelic "noise" that's being attached to the 5-HT receptors (where sensory input is also received), as well as forming new ones associated with the 'false' stimuli received from the drug. In doing so, this allows those connections to become activated again in the future.

Our existing pathways are why we can for instance see a red logo with some white script lettering, and recognize it as the "Coke" logo. The stimuli received from the eyes trigger a specific set of neural pathways to become activated which are associated with the memory which stores that set of information, and the red/white jargon is translated in your head as "Coke".

When you sense something which triggers a pathway created as a result of the false stimuli brought on by a psychedelic compound, your brain does the same thing, attempting to translate it into what it knows. In this case however, the result is quite a bit different. Essentially you are rewiring the brain to respond with memories of stimuli that never actually existed. This can result in a sense of increased awareness/sensitivity to slight differences as you mentioned, along with a slight distortion on your normal perception. (If severe enough, that can be classified as a disorder called HPPD, but for most people it's really mild, and they are not significantly bothered by this effect.)

TLDR: LSD made new neural connections based on fake stimuli from the drug, which have slightly changed the way your brain works.