The inputs in the skill coefficients section represent the combination of of each damaging skill and effect used by your character (as well as some effects/modifiers that are not input elsewhere in the optimizer). These inputs are usually reverse-engineered from ingame logs, but they can be changed manually to simulate different rotations, scenarios, equipment, etc.
While one could input the coefficient values for e.g. a single skill cast, resulting in the optimizer optimizing for and outputting the total damage dealt by that single skill, typically the skill coefficient inputs will be the averages over the entire duration of a rotation. This means the optimizer output will be the average DPS of that entire rotation.
Power coefficients are entered as the product of weapon strength * skill coefficient. For example, a geomancy sigil's effect has, according to the wiki, a coefficient of 0.25, and sigil effects have a weapon strength of 690.5. If it were triggered, for example, 5 times in a 60 second rotation, it would contribute:
0.25 * 690.5 * (5 / 60) = 14.39 average power coefficient
Condition coefficients are entered as the total number of seconds of condition they deal (stacks * duration). For example, a geomancy sigil's effect inflicts 3 stacks of bleeding for 8 seconds, or 24 total bleed ticks. If it were triggered 5 times in a 60 second rotation, it would contribute:
24 * (5 / 60) = 2.0 average bleeding
Summing every weapon and utility skill cast and every effect triggered in a rotation in this way should result in a good approximation of the total DPS output of the build.
There is a caveat, however:
As described in How It Works, some specific bonuses are not calculated by the optimizer, such as traits that buff skills of a certain type or only provide benefits some of the time.
Casting the scepter skill Grasping Dead, for example, normally has a power coefficient contribution of 0.8 * 1000 = 800 (with an ascended scepter) and a bleeding contribution of 3 * 10 = 30 stacks, but on a necromancer equipping the scepter-specific Lingering Curse trait, one would instead add 3 * 10 * 1.5 = 45 stacks of bleeding per cast. This cannot be done automatically by the optimizer because it does not know what portion of the total bleeding comes from each source.
Be careful of this when making comparisons using coefficients. For example, if comparing 3% sigils and hydromancy sigils on power soulbeast, be sure to account for the hydromancy sigil always triggering with "Sic 'Em!" and Honed Axes active.
Why are these values what they are?
The damage dealt by a skill can be broken down completely as:
Power Damage = (Weapon Strength * Coefficient / Target Armor) * Effective Power * Optimizer simulated modifiers * Optimizer ignored modifiers
Burning/bleeding/etc Damage = (Stacks Applied * Base Duration) * Damage Per Tick * Optimizer simulated modifiers * Optimizer ignored modifiers
Total DPS is calculated as the sum of these damages for every skill cast, divided by the total combat time.
These can be rearranged to separate out what the optimizer simulates, and what it does not:
Power Damage = (Effective Power * Optimizer simulated modifiers / Target Armor)
* (Weapon Strength * Coefficient * Optimizer ignored modifiers)
Power DPS = (optimizer stuff)
* average (Weapon Strength * Coefficient * Optimizer ignored modifiers) per second
Burning/bleeding/etc Damage = (Damage Per Tick * Optimizer simulated modifiers)
* (Stacks Applied * Base Duration * Optimizer ignored modifiers)
Burning/bleeding/etc DPS = (optimizer stuff)
* average (Stacks Applied * Base Duration * Optimizer ignored modifiers) per second
Thus, the skill coefficient data for power is "the average amount of skill coefficient * weapon strength my skills deal per second," and for burning it is "the average amount of stacks of burning * duration my skills deal per second" (both including specific modifiers).
In short, it is exactly how strong your build's rotation is, independent of gear or buffs.