後はこの仕掛けを偶然を装って作動させることで、彼に対して災難が降りかかる形。
roughly translated it's something like
afterwards by running the mechanism/device which is disguised as a accident, great disaster will fall upon him
させる is used to make/let someone do something.
However in this case the someone is the mechanism/device (which is a thing) so how does it work in this case?
後はこの仕掛けを偶然を装って作動させることで、彼に対して災難が降りかかる形。
First, 作動する is almost always intransitive, whereas English run is both transitive and intransitive. In English, you can safely say both "the system runs" and "run the system". In Japanese, you can say システムが作動する but usually not システムを作動する. (Actually, 作動する may be occasionally used transitively by some people, but it must be fairly uncommon. I checked the first 200 examples of 作動 on BCCWJ Corpus and found no example of 作動する used as a transitive verb.)
When you want to make a causative form, for most intransitive verbs, the agent (or "causee") must be marked with を. See this if you're not familiar with causation in Japanese.
So 仕掛けを作動させる in this sentence literally means "(I) make the device run/function", not "(I) run the device". The subject of the verb (作動させる) is the speaker, which is omitted in this sentence, as usual.
作動させる === Cause it to activate (deploy, operate, ...)
(後は...) この仕掛けを偶然を装って作動させることで、彼に対して災難が降りかかる(形。)
(All that's left to do is ... )
By causing the mechanism to activate (deploy, operate, ...) accidentally, (produce the pattern/result/appearance of) great disaster falling upon him
--- except it's a fake(d) accident
