Why is main hand cold damage 10-15? What exactly was enhanced by this support gem?
This happens because of your current skill tree. You gain 71% projectile damage from it.
Since your support gem is linked to a projectile skill it benefits from the bonus damage from your skill tree.
6*1.71 = 10.26, 9*1.71 = 15.39
It will also be affected by area damage or bow damage modifiers since it has that in the main skill gem description (the latter because you are using a bow).
You can see the skill descriptions and modifiers of every gem right below the name of the skill:
When it comes to determining whether it's a projectile or not you just have to watch the skill. Skills that look like projectiles but aren't are usually described in their attribute as "Melee" (Lacerate for example).
The cold portion of the spell will of course also be affected by increased elemental or cold damage.
You have 71% increased Projectile Damage from your skill tree. Caustic Arrow is an attack skill that fires projectiles, so the added cold damage would also be modified by that 71%.
Min Damage: 6 * (1 + .71) = 10.26
Max Damage: 9 * (1 + .71) = 15.39
You can also see this in your main physical damage from the bow.