Part of rule 117.10 says:
the resolution of a spell or ability doesn’t pay another spell or ability’s cost, even if part of its effect is doing the same thing the other cost asks for.
I am wondering if this means that the resolution of an ability can't be used to trigger another ability?
Yes, it can.
A simple example would be a "Tim" that taps for damage, like Prodigal Pyromancer. Combine that with any "Enrage" ability, like Bellowing Aegisaur. If Prodigal Pyromancer's ability targets Bellowing Aegisaur, then when it resolves, it will trigger Bellowing Aegisaur's ability.
Here's rule 117.10:
Each payment of a cost applies to only one spell, ability, or effect. For example, a player can’t sacrifice just one creature to activate the activated abilities of two permanents that each require sacrificing a creature as a cost. Also, the resolution of a spell or ability doesn’t pay another spell or ability’s cost, even if part of its effect is doing the same thing the other cost asks for.
What 117.10 is saying is that if an ability requires a cost such as "Sacrifice an artifact" (Throne of Geth), then sacrificing an artifact as the result of the effect of another spell or ability (Storm Fleet Arsonist) doesn't pay for that cost.
Or similarly, if you have an ability that requires you to pay life to activate it, losing life from another ability doesn't count as paying that cost.
Note that 117.10 is under the section on "costs", so it is really talking about activated abilities, not triggered abilities. Activated abilities occur due to a player choosing to pay a cost to activate them. Triggered abilities happen whenever something happens.
This is the difference between "Sacrifice a creature: gain 1 life" and "Whenever you sacrifice a creature, gain 1 life". The latter will trigger as a result of another ability making you sacrifice a creature, but the former will not.
See this often-duplicated question for a similar common confusion.
Activated abilities have costs that need to be paid. Triggered abilities have conditions that need to be met.
Here's all of 117.10 from the comprehensive rules:
117.10 Each payment of a cost applies to only one spell, ability, or effect. For example, a player can’t sacrifice just one creature to activate the activated abilities of two permanents that each require sacrificing a creature as a cost. Also, the resolution of a spell or ability doesn’t pay another spell or ability’s cost, even if part of its effect is doing the same thing the other cost asks for.
The resolution of another ability can be the triggering condition of a triggered ability, like say Day of Judgment's resolution destroying all creatures to trigger Deathgreeter, all those creatures die so you gain one life for each.
Where the issue is say you play Storm Fleet Arsonist after combat and your opponent has Mogg Fanatic. Your opponent needs to sacrifice something because of your Arsonist's raid ability, they can't sacrifice Mogg Fanatic to your Arsonist's ability and use that sacrifice to deal damage with Fanatic's own ability. If they use Fanatic to deal damage, they still need to sacrifice something else for Arsonist, if they sacrifice Fanatic for Arsonist, they can't use it's ability to deal damage.
One thing to keep in mind, if a spell or ability creates mana or tokens, those can be used to pay costs, but they become their own objects, mana in your mana pool, tokens on your battlefield, and it's not the resolution of the effect you are using to pay, just what that resolution created.
