How does LSD affect the brain and neurochemistry?

No one knows definitively. The best scientific explanation that I've seen is that LSD triggers a cascading release of serotonin and other neurotransmitters in the brain, resulting in heightened neural activity which the conscious mind then struggles to interpret sensibly (leading to altered perception, affect, and hallucinations).

An interesting side note, only around 3/4 of a percent of lysergic acids (LSA, ALD, LSD, ect.) actually cross the blood-brain barrier.

Some theorize LSD blocks serotonin receptors, while others believe LSD agonizes them. The most widely accepted theory is that LSD does both; blocking certain receptors and agonizing others. In a "layman" sense, the blockage and excitement of these receptors causes the brain to "skip a beat", and have to learn to regulate it's serotonin levels. While this is happening, the brain seems as if it is "re-wiring" itself, causing the user to face different levels of emotions and states of psychosis as the "trip" progresses. Depending on the mental status and environmental surroundings of the user, an LSD experience can either cause prolonged psychosis or a heightened sense of self awareness. How long these effects stay with the user depends on the intensity of the trip and how prepared the user was to the effects at the time of use.