This question actually originated from the games, but it applies to the anime/manga as well.
Water is strong against fire because water can extinguish fire. Electricity is weak against Grass because grass is grounded, which nullifies electricity.
What is the reasoning for Bug to be effective against Dark type?
And I've already heard about the Malaria in Africa joke, it's not funny!
If I had to guess, it was because of game balance more than logic.
There are only two types that are super-effective against Dark.
Neither of which make much logical sense. But it would be fairly safe to say that Dark would be overpowered if it only had one or no weaknesses.
Another argument to support the game-balance motivation is that bug types would otherwise be considered weak since they are super-effective against few other types.
So in short, they probably needed to keep Dark in check and make Bug more competitive.
This sort of balancing isn't particularly new. Psychic is a good example.
In the first generation of games, Psychic had only one weakness - Bug. Despite ghost also being super-effective against Psychic, there weren't any "real" ghost attacks in generation 1 (and only a couple of bug attacks).
So generation 2 added Dark types to keep Psychics in check.
Dark types are weak to (and resisted by) fighting because most types of fighting emphasize discipline; many of them put themselves in painful situations to strengthen themselves.
However, the dark type doesn't train much, and tend not to do well in fair combat because they are physically weaker than a given fighting type, and not tough enough to withstand them, either.
Now, as for bugs, one of their attributes is that they tend to be cooperative, and their attacks are nothing but direct to the goals they instinctually show. Dark types are super effective on psychic types because psychic types have the same indirect tricks, but the intimidation of dark types put fear into psychic types, and you can't really reason with a psychopath using psychology tricks or casting uncertainty or confusion into their minds because of their strong will. For bugs on dark, it's the same thing as fighting on dark. The dexterity and organization bugs naturally develop overpower the dark type's battle coordination, because their specialty is fighting unfair, but if a group of insects attack you, there's no backup plan one can use to get them off of you, short of maybe jumping into water, setting yourself on fire, or otherwise cloak yourself in something which keeps the bugs from continuing to attack you.
The Dark type is an embodiment of dirty tactics. The original Japanese translation is 'Evil' type, and this is shown as giving the Pokemon of this type underhanded or sneaky ways of fighting. Examples of this are shown in various Dark type moves:
Pursuit : Dealing more damage on an enemy trying to retreat
Sucker Punch : Attacking before the enemy is ready
Fake Tears: Pretending to cry to put an enemy off guard
Beat up: Attacking with the whole team in an unfair 6v1 fight
Culturally in Japan, the villainous dark character must be beaten by the virtuous hero character. That is why Fighting types are super effective. They embody this heroism trait.
So why also Bugs? This is probably a reference to the series Kamen Rider, a show that has existed continuously from the 1970s to the present day and is a cultural phenomenon large enough that every child in Japan would know it. In this series, the usual premise is a Masked Hero riding a motorcycle fights the forces of evil with the Main character always using an Insect themed costume.
In fact, the Bug type move Signal Beam(シグナルビーム) is pronounced as English in the original Japanese versions, reminiscent of many Kamen Rider 'special moves'.
This insect association with heroism actually goes back further, with stag beetles representing the honourable samurai and becoming the basis for many real samurai helmet designs.
Another hint towards this connection exists before the Dark type was ever introduced. In generation 1, the villains often used Poison types. This is the only Generation where Bug types were super effective against poison!
Dark was 100% a balance to Psychic in Gen II, Psychic dominated the typing tree and what little multiplayer competition there was during Gen 1 (this was also due to the fact that there was no separate "Special Attack" and "Special Defense" in Gen 1, only "Special" which covered both and made Psychics who normally excelled in the stat both powerful AND bulky).
BUT one can see reasoning behind the chosen typing beyond convenience of balancing the types.
A key thing to note is that the Japanese name for the "Dark" type is more closely translated to "Evil" or "Sinister." This is why dark types have many underhanded or villainous move names like "Quash," "Sucker Punch," "Fake Tears," "Torment," and "Nasty Plot." Also why the Fairy type, which embodies goodness or purity, has a type advantage.
Alternatively, Fighting type embodies heroism or, as Beetle said, discipline. If you consider the culture of classic Japanese action films, the heroes who take down EVIL are usually adept and disciplined in martial arts.
On the Bug side, insects obviously aren't know to be the undoing of evil, but if you consider what people associate with bugs, things become more clear:
So ultimately, Evil (Dark) is framed as being bested by discipline/heroism (Fighting), the collective effort of the masses (Bug), or pure goodness (Fairy).
Now if only somebody could explain why exactly it is that Ghost type is weak against Dark type...
The answer is simple: Kamen Rider.
Kamen Rider was (and probably still is) an insanely popular Power-Rangers style action show in Japan where the hero is insect-themed. An insect themed hero (as "Goosh" mentioned vaguely) who fights against the forces of EVIL (Dark Types).
So you can bet Bug > Dark is purely the result of Kamen Rider.
