I think it's about time that I post the first section of this answer. I believe I can only fit 2 complete types into one answer (Normal and Fire), so that's what I'll do. I'm not done with the rest of the matchups, but this is going to be broken into various answers anyway.
Normal vs Normal- normal “people” (I know normal types are usually animals, but I’ll just compare the typing to your average everyday person) tend to just walk by each other, not really interacting with each other. This would explain the neutrality.
Normal vs Fire- while fires cause mass destruction to everyday people, people have adapted to things catching fire, and usually people react before the situation gets out of hand.
Normal vs Water- water usually can’t harm the average person, unless a natural disaster like a tsunami or hurricane occurs. The opposite is also true, people/animals can only hurt water via pollution.
Normal vs Electric- it’s very uncommon for a human/animal to be hit by lighting, seeing as lighting hits the object closest to the cloud (usually). Of course, electric shocks can be deadly.
Normal vs Grass- while people and animals are more than capable of destroying plants, they usually help plants grow. Also, plants help animals and people survive.
Normal vs Ice- people and ice barely interact, that’s really it.
Normal vs Fighting- fighters are trained to, well, fight better than the average person, so this is a pretty simple and logical type match up.
Normal vs Poison- while poison kills humans/animals, it isn’t very common that they come in contact. Plus, Poisoning can be cured using antidotes, so it’s not as huge of a threat (obviously poison/venom is still very dangerous, it can just be cured).
Normal vs Ground- the ground supports life, really the only way the ground can harm the average being is via earthquake/quicksand.
Normal vs Flying- again, there isn’t much of an interaction here. I will state that a lot of Flying types are part Normal type.
Normal vs Psychic- people seek psychics for help with their problems, and average people don’t have the power needed to beat a psychic in combat, so they remain neutral.
Normal vs Bug- most bugs are pretty harmless, and people tend to ignore them. So, again, there isn’t much interaction between them.
Normal vs Rock- rocks are very sturdy, and most rocks/minerals won’t break easily via contact from a human/animal. Normal people just don’t have that kind of strength, so it’s pretty reasonable why Rock resists Normal.
Normal vs Ghost- this is a pretty simple matchup. Beings can’t touch ghosts, so the opposite is (technically) automatically true, meaning both are immune to each other.
Normal vs Dragon- not much of an interaction here, and while dragons could cause mass chaos to the everyday population, I guess dragons wouldn’t be a huge threat due to the sheer amount of people and animals.
Normal vs Dark- “Dark” translates to “Evil”. While “evil” people can cause destruction and harm to normal people, when it all boils down, the good guys win. Normal people, of course, are not considered “heros” or “the good guys”; so Dark vs Normal is neutral.
Normal vs Steel- the same concept of Rock resisting Normal applies to here. Normal people just can’t break a piece of metal, so Steel resisting Normal is very reasonable.
Normal vs Fairy- not much interaction here, a mythical creature and a normal being. Though the same could be said for Dragon vs Normal.
Fire vs Fire- fighting fire with fire almost never works in any party’s favor, plus, it is part of the starter type’s type chart to resist itself.
Fire vs Water- most fires are put out by water, so it makes a lot of sense for Fire to be weak to Water. And, while Fire can technically evaporate Water (well, heat can), it Water will still counteract most fires before they have a chance to fully evaporate water.
Fire vs Electric- Electricity causes fires, so they sort of link together. The interaction between them doesn’t really exist though.
Fire vs Grass- [dead] plants help fires spread, as they easily catch on fire. A fire can also burn plants, so it’s very logical why Fire is super effective against Grass. Also, plants only make a fire stronger, explaining Fire’s resistance to Grass.
Fire vs Ice- fire (heat, but fires emit heat) will melt ice when the ice is a) set on fire b) near a fire. While ice will turn into water when melted, I don’t really think that’s enough to say that this type matchup doesn’t make sense.
Fire vs Fighting- you can’t fight a fire with physical strength, but they have little to nothing to do with each other besides that.
Fire vs Poison- while people believe that heat can get rid of toxins, I’m not sure how scientific that is.
Fire vs Ground- dirt can be used to put out fires, and dirt is part of the “ground”.
Fire vs Flying- fires can’t really touch birds, however, if a bird were to come into contact with a fire, it would not end well… It makes a good dinner though.
Fire vs Psychic- Psychic is often associated the mind (for example why Bug is super effective against Psychic, which I’ll cover later), so you could say that people are, to a certain extent, scared of fire. On the other hand, Psychic could also be compared to magic, which can generate fires, so their interactions slightly counteract each other.
Fire vs Bug- fires destroy plants and animals, including helpless bugs. If you’ve ever heard of cooked bugs, such as crickets, well, some of them are cooked with a fire.
Fire vs Rock- very similar to Fire vs Ground, sand can be used to put out fires. Sand is rock that has undergone corrosion.
Fire vs Ghost- again, I don’t see much of a connection/interaction between ghosts and fires. Quite a few ghost types learn Will-O-Wisp, but that feels slightly irrelevant.
Fire vs Dragon- dragons are powerful beasts with scaly skin that I assume is fairly tough. Fire wouldn’t really be very effective in those conditions. The stereotypical dragon also breathes fire, so they have likely adapted to, well, fire coming out of their mouths.
Fire vs Dark- “evil” people may cause fires, but like most of the types on this list, there isn’t really a clear cut interaction between them.
Fire vs Steel- Fire/heat melt metals, and steel types are (mostly) made of metal, or are part metal (I can only think of Lucario off the top of my head that’s not partly made of metal).
Fire vs Fairy- “I believe that fire can't be stopped by fairy because fire is a primitive force. One that has existed much longer then fairies and is with this a mystic and powerful form of magic. Fire is also a type of magic that can't simply be stopped by a common spell as it can scorch the spell into oblivion. In order to hit bewitched fire a fairy must summon rain or water and with this attempt to quench the fire. The need to summon water can be associated with the simple fact that they are not capable of doing the job themselves” source, I believe that this is correct for Fire resisting Fairy, and I’ll probably quote this answer a few times.
If I can fit Water into this answer once I'm done with all of its matchups, I will.