I always thought bugs were strong against psychic because they had underdeveloped brains that were harder to influence.
Similarly, dark and ghost have stronger roots in the supernatural, as well as that psychological edge, and are thus naturally good at fighting a supernatural power. It's worth noting dark is "evil" and not an absence of light.
Ground beats steel because steel pokemon are heavier and rely on a solid ground underneath them. Also think about steel in an earthquake (or even a tuning fork)
Psychic beats poison because psychic's strength lies in the mind, rather than the body, and is thus less susceptible to poison.
Flying is strong against grass (and bugs) because birds eat plants/seeds/bugs and are adept at hunting and scavenging in/from/around plants.
Bug is strong against dark because dark holds no psychological edge over their underdeveloped brains (what do they care if something looks evil?)
Fighting beats dark because they are focused and can fight through fear.
Fighting beats normal cause its literally the stronger version of normal (I always imagine a fighter beating up a normal person, or a pillow)