Creative works are often hard to describe quickly and effectively so writers often use comparisons or "They Fight Crime!" in order to get their point across. However, despite constantly reminding us that Drugs Are Bad, writers have a strong addiction to them.
This is a trope for works which are forever and eternally described as being on some kind of mind-altering substance. The most common ones are crack (highly concentrated cocaine), acid (LSD), and steroids, but any drug can and will be used. It's often used for Mind Screw stuff, but by no means exclusive to them.
The choice of drug can imply certain qualities. Works on speed are extremely energetic and fast-paced; works on crack are similar but with an element of surrealism or wackiness implied. Acid and weed are more psychedelic and confusing, whereas works on steroids are usually bigger and more extreme. One would imagine a work "on NyQuil" would be dreamy and lethargic, while describing Show B as Show A "on its meds" implies that Show A is in some way weird, while Show B, while cut from the same cloth, is less-weird. Finally, a work on "angel dust" (PCP) would most likely be extremely violent, angry, or megalomaniacal in tone.
Note that the chances are the creators weren't actually on drugs at all. No matter how slim those chances might seem.
The name of this trope came from an American anti-drug commercial from the Partnership To End Addiction, which showed an egg with the voice-over "This is your brain," then the egg being cracked on the side of a frying pan and the contents poured in, with the voice-over "This is your brain on drugs. Any questions?" This got so parodied that for a while people ordering eggs sunny side up would say, "I'll take a brain on drugs with toast," for example. (It might also explain why eggs over easy were so common for a while.) For an extra bit of trivia, the girl in the second version of the commercial? That's Tifa years later, among other people.