Because magic?
This really isn't a rules question.
I would guess that the reason in, when those spells were designed, the person doing though thought transformation was the best fit. Whether that is true or not may be debatable, but that is the reason.
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This is a game design question, not rules. Flagged.
As to why, flame arrow is grandfathered in from 3.5. The others I'd be happy to look at with links.
School selection can be promblematic. Generally, evocation spells call energy into existence and apply it. Conjugation brings items into existence. Transmutation changes things. Flame arrow doesn't directly effect the target of the arrow. Rather, it transforms the arrow, which then affects the target of the shot. This sort of intermediary item is almost always transmutation.
You've described flame arrow as adding fire rather than being changed. The arrow is changed bit now burns when fired.
See if the same sort of distinctions in the other spells.
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The schools are poorly defined and overlap a lot.
Transmutation seems like the school they put a lot of the "this fits in multiple schools" and "this fits in none of the schools perfectly" spells. Mostly because this:
Transmutation spells change the properties of some creature, thing, or condition.
is a ridiculously broad description which could encompass nearly every spell.
Along with all of the transformation spells, this makes the list long indeed (although the sheer number of X Shape/Physique/Anatomy Y spells makes this somewhat artificial).
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I blame Kaer Maga.
On a serious note, the main reasons were already listed. Grandfathered spells from previous editions and an extremely broad description. Transmutation tends to be the dumping ground for spells which don't fit neatly into a single other school.
1st edition wrote:
Level: 3 Components: V, S, M Range: Touch Casting Time: 3 segments Duration: 1 segment/level Saving Throw: None Area of Effect: Each arrow/bolt touched
Explanation/Description: Once the magic-user has cast this spell. he or she is able to touch one arrow or crossbow bolt (quarrel) per segment for the duration of the Flame Arrow. Each such missile so touched becomes magic, although it gains no bonuses "to hit". Each such missile must be discharged within 1 round, for after that period flame consumes it entirely, and the magic is lost. Fiery missiles will certainly have normal probabilities of causing combustion. and any creature subject to additional fire damage will suffer +1 hit point of damage from any flame arrow which hits it. The material components for this spell are a drop of oil and a small piece of flint.
2nd edition wrote
(Conjuration/Summoning)
Range: 30 yds. + 10 yds./level Components: V, S, M
Duration: 1 rd Casting Time: 3
Area of Effect: Special Saving Throw: None
This spell has two effects. First, the wizard can cause normal arrows or crossbow bolts to become magical flaming missiles for one round. The missiles must be nocked and drawn (or cocked) at the completion of the spell. If they are not loosed within one round, they are consumed by the magic. For every five levels the caster has achieved, up to 10 arrows or bolts can be affected. The arrows inflict normal damage, plus 1 point of fire damage to any target struck. They may also cause incendiary damage. This version of the spell is used most often in large battles.
The second version of this spell enables the caster to hurl fiery bolts at opponents within range. Each bolt inflicts 1d6 points of piercing damage, plus 4d6 points of fire damage. Only half the fire damage is inflicted if the creature struck successfully saves vs. spell. The caster receives one bolt for every five experience levels (two bolts at 10th level, three at 15th level, etc.). Bolts must be used on creatures within 20 yards of each other and in front of the wizard.
The material components for this spell are a drop of oil and a small piece of flint.
I always preferred this, with the 2nd version. At high level, it was much better than Fireball (which became capped at 10d6 in this edition).
Anyway, the reason for this spell being in Transmutation is not because it was grandfathered in from the earlier editions. 3.5 lists it as Transmutation, so the best answer is 'A wizard (of the coast) did it'.
Conjuration is really massive too. I've often considered bringing back earlier edition's 'automatically opposed school' for Specialist Wizards [where the opposed school could not be cast from in any way] and matching Conjuration and Transmutation up against eachother.
Might even leave free choice of opposed school to the rest of the schools.
If you did that then wizards would just continue to pick Divination and use the really powerful spells in the Conjuration and Transmutation lists with impunity.
I'm not sure if balance was what you were trying to go for with that idea, but if it was it's not going to work.
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See, doing that means they aren't specializing in Transmutation and Conjuration for all the eternally useful bonus spells.
Now sure, Divination spells aren't total garbage but they aren't the sort of power houses those other two schools are pretty full of.
Granted, Divination does have rock solid powers associated with it...
Nothing prevents you from taking Spell Focus (Conjuration) or (Transmutation) if you are a Diviner wizard, but they have the best school powers by far, with ungodly initiatives, ability to always act in a surprise round, and a decent buff for allies at low levels when they don't have enough spell slots to last the adventuring day.