The ring of mystic fire only affects the upcoming spell once

The ring of mystic fire (Magic Item Compendium 125) (7,500 gp; 0 lbs.) does, indeed, grant "a bonus to the amount of damage you deal with the next fire spell you cast before the end of your turn," but the ring only grants that bonus once. Each ray of a scorching ray spell isn't the spell; instead, the whole spell creates multiple rays. The ring improves the spell's damage only once by a maximum of +4d6 no matter how many effects result from the ring-modified spell.

(Ruling that each ray of the scorching ray spell benefits from the ring multiplies the power of the ring by the number of rays—with rather extreme consequences at low levels, especially given that the ring increases also the wearer's caster level for fire spells!)

Although the DM has several options available to adjudicate the ring's effect—allowing the player to distribute the total number of extra dice damage the ring grants among the rays before (or even after) the rays're launched, for instance—, this DM would keep things simple and have the player add the extra damage to his PC's first ray in a volley from the spell scorching ray, in much the same way that precision damage applies to the first attack in a similar volley (Rules Compendium 42). (In fact, I have a fire wizard in a current campaign who's eyeing this very item, and that's how I'm ruling it works.)

Hence, for example, under such a ruling, a level 6 caster wearing a ring of mystic fire can take a swift action to spend 3 of the ring's charges. Then the caster can cast scorching ray, creating 2 rays (because of the increase in caster level granted by the ring) and making 2 ranged touch attacks. If the first ray hits, it deals 8d6 points of fire damage. If the second ray hits, it deals 4d6 points of fire damage.

Add your spell attack bonus to each attack roll.

When a spell tells you to make an attack roll, that means you roll a d20 and add some bonus to get the final value. If that final value matches or exceeds the target's AC (Armor Class), then the attack was successful, otherwise it was not. Chapter 10 of the Player's Handbook defines this term in the subsection "Attack Rolls", and says the following:

Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

And the text of Scorching Ray (Chapter 11) says that the caster should make an individual ranged spell attack roll for each ray:

You create three rays of fire and hurl them at targets within range. You can hurl them at one target or several.

Make a ranged spell attack for each ray. On a hit, the target takes 2d6 fire damage.

For each ray, you roll a d20 and add the spell attack bonus to the roll, and determine whether that ray hit your intended target. If a ray hits, then roll to calculate the damage from that ray. Attack rolls and damage rolls are independent from each other, and in the case of Scorching Ray, each ray's damage is 2d6 (i.e. sum of 1d6+1d6).